Interaction of a turbulent spot with a two-dimensional cavity

Wu, S.; Yaras, M. I.

doi:10.1063/5.0062617

A study with experimental and numerical components is conducted to establish the effect of a two-dimensional surface cavity on the structure and growth rate of a turbulent spot. The spot is artificially created in a two-dimensional laminar boundary layer developing under zero streamwise pressure gradient. Interactions of the turbulent spot and its wake with the Kelvin–Helmholtz rollers of the cavity shear layer are shown to result in significant increases in its lateral and streamwise growth rates. The underlying physics of these developments are identified.

I. INTRODUCTION

A. Physics of turbulent spots in boundary layers

A typical turbulent spot contains multiple generations of wavepackets of coherent vortical structures.^1–3 Mutual interactions among the wavepackets produce a seemingly random pattern of vortical structures within the spot. However, upon formation, a wavepacket tends to consist of a relatively more organized set of approximately hairpin-shaped vortical structures.^1,4,5 The vortices tend to approximately align in the streamwise direction in groups of three to five, with each hairpin vortex approximately oriented with its streamwise legs extending in the upstream direction and its head lifted away from the solid surface due to self-induction effects.^1,4 The flow physics underlying the formation of such a wavepacket are more readily traceable along the side edges of a turbulent spot. At these locations, hairpin vortices of prevailing wavepackets locally destabilize the laminar flow field adjacent to the turbulent spot, thereby promoting the formation of new wavepackets.^1,6,7 The typical process of destabilization is realized by the formation of streamwise streaks of velocity perturbations, induced by the upwash and downwash motions of the streamwise legs of prevailing mature hairpin vortices, schematically illustrated in Fig. 1. These streaks adopt a predictable spanwise spacing dictated by a balance between increased viscous diffusion in small-wavelength perturbations and decreased wall-normal induced velocity in large-wavelength perturbations.⁸ The resultant spanwise spacing has a mean value of 100 viscous wall units, with a lognormal probability distribution with a dense range between 60 and 180 viscous wall units.^4,9,10 The viscous wall unit is based on the kinematic viscosity, ν, density, ρ, and wall shear stress, τ_w, through $ν / \sqrt{τ_{w} / ρ}$ ⁠. The shear layer enveloping each such velocity streak is known to be inviscid unstable to sinuous and varicose modes,^11,12 resulting in streamwise grouping of the vorticity of the shear layer around the streak at predictable wavelengths of several hundred viscous wall units.^4,13 These groups of vorticity subsequently develop coherence to form new hairpin vortices as illustrated in Fig. 1, which shows one of a multitude of spatial arrangements where they straddle the streamwise legs of the parent hairpin vortex.^14,15 The same fundamental mechanism of new wavepacket formation likely also prevails within the turbulent spot.¹ Albeit, the cause-and-effect relations in the flow development are notably more difficult to discern in these regions in mature spots as multiple wavepackets of different generations, sizes, and states of partial/full overlap participate in the process. The noted process of wavepacket formation along the sides of a turbulent spot has been suggested to be a primary mechanism driving the spanwise growth of a turbulent spot.¹ While the rate of this growth is shown to be independent of the original disturbance creating the spot,¹⁶ it is also found to negatively correlate with streamwise acceleration of the local freestream.^17–19

FIG. 1.

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Idealized schematic of a parent hairpin vortex straddled by two younger wavepackets.

Younger generations of wavepackets in the spot are located in closer proximity of the solid surface where they form and convect in the streamwise direction at relatively low velocity.¹ The older wavepackets are larger, extending as far as several local undisturbed boundary layer thicknesses away from the surface, and accordingly convect downstream at higher velocity.^16,20 Expectedly, this results in the larger wavepackets to preferentially locate near the leading side of the spot, and the spot trailing edge to be populated by younger-generation wavepackets. The noted difference in the leading and trailing edge convection speeds is a primary mechanism driving the streamwise growth of the turbulent spot.^17,20 The heads of the larger hairpin vortices at the spot leading edge extend over the laminar boundary layer ahead of the spot and promote entrainment of laminar fluid, which further promotes the growth of the spot.^16,20

The turbulent spot is well-known to promote stability or calming effect in its wake, an effect realized by perturbation spanwise vorticity trailing the spot in close proximity of the surface. The downwash induced by this vorticity yields a relatively fuller and thus more stable wall-normal profile of streamwise velocity.^1,21,22 Viscous diffusion of this vorticity ensures the streamwise length of the calmed wake to be finite.¹ This effect of the perturbation vorticity is reduced under the influence of favorable streamwise pressure gradients.²¹

B. Physics of the shear layer over a two-dimensional rectangular cavity

In broad terms, the shear layer over a two-dimensional cavity behaves in a manner similar to a free shear layer, with the Kelvin–Helmholtz instability mode promoting streamwise-periodic amalgamation of spanwise vorticity, at times leading to the formation of spanwise vortices (K–H rollers).^23–25 The length scales deemed relevant to this shear layer development are cavity length (L), cavity depth (D), and the boundary layer momentum thickness at the cavity upstream edge (θ_s).²⁶ The streamwise motion within the cavity driven by the shear layer is terminated by the downstream bounding wall of the cavity, resulting in the formation of a primary vortex in the cavity, which scales on the shorter of the L and D dimensions.²⁷ Upon impingement on the downstream edge of the cavity, parts of the K–H rollers are clipped and merged with the recirculating flow within the cavity.^28,29 The pressure fluctuation generated by the impingement process at the dominant K–H frequency feeds back into the evolution of the cavity shear layer, thus creating a feedback loop to produce notably more distinctive peaks in the pressure and velocity spectra of the cavity shear layer than in an unconfined free shear layer.^30,31 The natural frequency of the cavity shear layer, expressed as a Strouhal number, ${St}_{θ_{s}} = f θ_{s} / U_{0}$ ⁠, is within the band of frequencies amplified by the Kelvin–Helmholtz instability and varies with $L / θ_{s}$ as shown in Fig. 2. Three distinct modes are evident, and the ${St}_{θ_{s}}$ value is noted to decrease with cavity length within each mode. Feedback-driven large-amplitude oscillations are generally not observed for $L / θ_{s}$ values below about 50.^27,32 This is attributed to the fact that rollers in these cases do not have enough time to grow to sufficient strength prior to impingement.^27,32 At times, feedback-driven oscillations and the regular formation of rollers within the cavity shear layers are not observed even for $L / θ_{s}$ values above this threshold.^33–35

FIG. 2.

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Variation of the dominant Strouhal number with cavity length for laminar, incompressible conditions: ■ ▲ ● Sarohia;²⁷ ◓ Knisely and Rockwell;³⁶ ◻ △ ○ Basley et al.;³⁷ × Direct numerical simulation from the present study; + Experiment measurements from the present study.

In some cases where the cavity shear layer originates from a laminar upstream boundary layer, the cavity shear layer may undergo secondary instabilities that result in the spanwise-wavy deformation of the K–H rollers prior to impingement.^38,39 In such cases, upon impingement the rollers are observed to reorganize into spanwise arrays of hairpin vortices.³⁹ The detailed mechanisms dictating the development of the vorticity field in such instances remain unexplored. In cases where the cavity shear layer originates from a turbulent upstream boundary layer, the formation of spanwise coherent rollers over a cavity tends to be inhibited by spanwise nonuniform deflections of the vorticity sheet induced by hairpin vortices prevailing in the upstream turbulence.³⁹ Despite this, clusters of small-scale coherent structures are observed to form in a streamwise-periodic manner at a Strouhal number of ${St}_{θ_{s}} \approx 0.016$ ⁠.^40,41 Streamwise streaks associated with upstream turbulence are also observed to lose their streamwise coherence beyond the first 1/3rd of the cavity length, which is attributed to the mutual interaction between the newly formed vortices within the cavity shear layer and the streaks in the prevailing turbulence.^39,42

Few studies^43,44 have investigated the effects of a cavity on triggering transition onset in a laminar boundary layer. A cavity appears to trigger transition by amplifying the Tollmien–Schlichting waves prevailing in the approaching boundary layer as well as through instability modes that develop within the cavity.⁴⁴ These effects appear to depend on both L and θ_s.⁴⁴

The present study aims to complement existing literature by investigating the effects of a cavity on a boundary layer, which is already in transition upon encountering the cavity. Specifically, the study aims to establish the extent to which the rate of transition is modified by such an encounter and the underlying physics of this interaction. Such an understanding may then enable optimization of the streamwise location and geometry of the cavity to promote an effect on the transition process, which is favorable to the application involved.

II. EXPERIMENTAL SETUP AND DATA PROCESSING

The experiments were performed in a closed-circuit wind tunnel with test section dimensions of 508 mm height, 762 mm width, and 1900 mm length. A 19 mm-thick test surface with an elliptic leading edge of 3-to-1 axis ratio and a spanwise-oriented cavity forms the floor of the test section (Fig. 3). Some of the wind tunnel air entering the test section is allowed to escape under the test surface such that the flow at the test surface leading edge is at a nominal angle of attack of zero degrees. The sidewalls and ceiling of the test section are streamwise slotted. The slot dimensions were optimized as per established literature on the design of slotted wall wind tunnel test sections.^45–48 The use of slotted walls ensures zero streamwise pressure gradient in the test section.

FIG. 3.

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Cross section schematic of the test section.

The spanwise-oriented cavity on the test surface is $D = 527 mm$ deep and spans the full width of the test section. The upstream edge of the cavity is positioned $486 mm$ from the test surface leading edge. Three cavity lengths of 42, 49, and $63 mm$ are investigated. They, respectively, correspond to 46.5, 54.4, and 70.0 times the momentum thickness of the undisturbed laminar boundary layer at the cavity upstream edge (θ_s), and to 33.8, 39.5, and 50.8 times the displacement thickness of the undisturbed laminar boundary layer at the location where the turbulent spot is generated (⁠ $δ_{0}^{*}$ ⁠).

A wall-normal jet of air is introduced into the test section flow intermittently through a pinhole of $0.75 mm$ diameter on the test surface located $179 mm$ from the test surface leading edge and offset by $15 mm$ from midspan. In each instance, the jet is activated for a period of $50 ms$ with a discharge velocity, which is 7.5 times the test section freestream velocity. A train of turbulent spots is produced by activating the jet at a frequency in the range of 1.1– $1.9 Hz$ ⁠, which was established to be suitable for allowing the laminar boundary layer and the cavity flow to return to their undisturbed states before the arrival of the next turbulent spot.

Measurements of the flow velocity as per the matrix given in Table I were performed with a single-sensor constant-temperature hot wire probe with a tungsten sensor of $1.3 mm$ length and $5 μ m$ diameter. The sensor is oriented in the spanwise (z) direction, which makes it sensitive to the x- and y-components of velocity. The body of the probe is inclined at an angle of $10 °$ to the test surface to best mitigate any flow blockage effects in the near-wall region.^49,50 The anemometer signal is sampled at a rate of $10 kHz$ and is low-pass filtered with a nine-pole filter of $5 kHz$ cutoff frequency, which is conservatively higher than the frequency range of the turbulence in the present flow. Sampling of the probe signal is synchronized with the time-periodic production of turbulent spots, and the data sampling duration is set to allow for the turbulent spot and its wake region to move past the probe. At each measurement position, 70 such spot generation and measurement cycles are performed, which was confirmed to be sufficient for statistical convergence of the ensemble-averaged velocity and the root mean square (rms) value of its fluctuating component.

TABLE I.

Measurement grid.

Streamwise locations of measurement planes	x = 328, 473, 530, 672 mm^a^, ^b
Wall-normal traverses per measurement plane	24 y-traverses from $z = - 80$ mm to z = 8 mm at increments of $Δ z = 3.8$ mm^c
Measurement points per wall-normal traverse	75 points from y = 0.2 mm to y = 16 mm clustered toward the wall
Sensor position uncertainty	x: ±1.2 mm; y: ±0.02 mm; z: ±0.6 mm

Streamwise locations of measurement planes	x = 328, 473, 530, 672 mm^a^, ^b
Wall-normal traverses per measurement plane	24 y-traverses from $z = - 80$ mm to z = 8 mm at increments of $Δ z = 3.8$ mm^c
Measurement points per wall-normal traverse	75 points from y = 0.2 mm to y = 16 mm clustered toward the wall
Sensor position uncertainty	x: ±1.2 mm; y: ±0.02 mm; z: ±0.6 mm

^a

Additional selective measurements were performed at $x = 400$ and 595 mm to capture the spanwise growth of the turbulent spot.

^b

Stated coordinates correspond to a cavity length of $L = 42 mm$ ⁠. When the cavity length is increased, x coordinates of measurement planes located at x > 522 mm are increased by the increase in cavity length.

^c

Additional traverses were performed at increments of $Δ z = 2.5$ mm to better capture the streaky structures in the spot.

The uncertainty in the measured instantaneous velocity is conservatively estimated to be $\pm 5 %$ ⁠, which increases to $\pm 9 %$ for flow velocity values less than $1.5 m / s$ ⁠. The stated uncertainty values are primarily dictated by the analog drift in the anemometer circuitry between calibrations. Compensation of the hotwire anemometer signal for ambient temperature variations during the experiment is based on the method of Bruun.⁵¹

The normalized perturbation velocity, $u'$ ⁠, is calculated as

u' (x, y, z, t) = \frac{u (x, y, z, t) - u_{l} (x, y, z)}{U_{0}},

(1)

where u is the instantaneous measured velocity, u_l, the “laminar” velocity, is the time average of the velocity measured at the location in the absence of the turbulent spot and its wake, and U₀ is the freestream velocity. The ensemble average of $u'$ and u over N = 70 measurement cycles are denoted by $\tilde{u'}$ and $\tilde{u}$ ⁠, respectively. During the experiments, the freestream velocity, U₀ was correlated with the pressure difference of the wind tunnel contraction leading to the test section which was sampled in conjunction with the hotwire signal.

The root mean square of the velocity fluctuation is calculated as

u_{rms} (x, y, z) = \frac{1}{U_{0}} \sqrt{\frac{1}{n} [\sum_{n} {(u (x, y, z, t) - \tilde{u} (x, y, z))}^{2}]} .

(2)

To determine the spatial extent of the turbulent spot, the leading, trailing, and spanwise edges are located at each cross-stream (y-z) measurement plane. It was found that the leading edge is most reliably identified by a specific value of the local perturbation velocity. Varying the threshold value of $u'$ in the 2% to 5% range yields leading edge locations that remain within 30 wall units distance of each other. This range is deemed sufficiently low to declare $u'$ a reliable indicator of the leading edge location. The final value of the location is based on the average of the locations corresponding to five evenly spaced threshold values of $u'$ between 2% and 5%.

As the wake region of the spot is not free of velocity fluctuations, $u_{rms}$ is found to be more effective than $u'$ in locating the spot trailing edge. This finding is consistent with published literature^20,22 and is also adopted for locating the spanwise edges of the spot in the present study. Locations corresponding to five evenly spaced threshold values of $u_{rms}$ are averaged in each instance. A threshold range of 4% to 8% was deemed most effective upstream of the cavity and 10% to 15% downstream of the cavity.

Normalization of all distance measurements with the viscous length scale is performed based on the spatially averaged wall shear stress under the footprint of the turbulent spot in the absence of a surface cavity. The instant chosen for this evaluation corresponds to the most mature state of the turbulent spot in the numerical simulation, with the spot leading edge located at $x / δ_{0}^{*} = 400$ ⁠. Data from the numerical simulation were preferred over the experimental data for obtaining the wall shear stress under the footprint of the spot owing to the higher spatial resolution in the numerical simulation. The wall shear stress extracted from the numerical data is within 20% of the same extracted from the wall-normal profiles of time-averaged streamwise velocity using the Clauser plot technique.

III. COMPUTATIONAL MODELS

A. Modeling in the Eulerian reference frame

The experimental investigation is complemented by a direct numerical simulation with consistent geometry and flow conditions. For computational efficiency, the generation of the turbulent spot via a wall-normal jet pulse and the downstream development of the turbulent spot are, respectively, captured in two separate simulations. The streamwise development of the spot is simulated with and without the cavity on the test surface to comparatively establish the effects of the cavity on the spot structure. The simulation of the spot generation (Simulation 1) and the simulation of the streamwise spot development in the absence of the surface cavity (Simulation 2) were previously performed by Brinkerhoff and Yaras.¹ Simulation 3 involves the presence of the surface cavity and was performed as part of the present study.

1. Computational domain and boundary conditions

The computational domains of the three simulations are shown in Fig. 4. Each domain consists of a flat, no-slip test surface at the bottom of the domain and free-slip surfaces at the sides and the top of the domain. A turbulent spot is generated in Simulation 1 at midspan with a pulsed jet from a rectangular subdomain serving as a jet pipe positioned $124 δ_{0}^{*}$ downstream of the leading edge of the no-slip surface, where $δ_{0}^{*}$ is the displacement thickness of the undisturbed boundary layer at the jet location. The domains for Simulations 2 and 3 are each larger than for Simulation 1 to allow for the growth of the spot without artificial influence from the free-slip surface boundary conditions. The free-slip condition allows for motion only within the plane of the boundary. Avoiding any artificial influence by this constraint on the local velocity magnitude and direction requires that the boundary be placed in a region of flow where the natural flow motion is consistent with this constraint. The spanwise boundaries are placed such that they are at least $30 δ_{0}^{*}$ away from the closest edge of the turbulent spot at its most mature state in the respective simulations. Insensitivity of the spot development to the spanwise domain width was confirmed by Brinkerhoff and Yaras,¹ who observed no change in the spanwise growth rate of the turbulent spot with 50% increase in the domain width of Simulation 2. The domain height for each simulation is adjusted in the streamwise direction to follow the displacement thickness distribution of a Blasius boundary layer at the local flow Reynolds number. This ensured a constant freestream static pressure along the length of the domain.

FIG. 4.

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Schematic of the computational domains: (a) Simulation 1: turbulent spot initialization; (b) Simulation 2: turbulent spot development without a surface cavity; (c) Simulation 3: turbulent spot development with a surface cavity.

The upstream edge of the cavity is $274 δ_{0}^{*}$ downstream of the spot-generating jet. This distance is sufficient for the spot to develop several generations of wavepackets, thereby developing an internal structure, which is representative of more mature states of turbulence.¹ The cavity has a streamwise dimension of $L = 30 δ_{0}^{*} = 46.5 θ_{s}$ ⁠, and a depth of $D = 300 δ_{0}^{*} = 465 θ_{s}$ ⁠. The depth of the cavity is chosen to ensure that the spot is not influenced by the bottom wall of the cavity. The cavity spans the entire domain which corresponds to more than seven cavity lengths. This span is notably larger than that required to avoid three-dimensional flow in the cavity due to end effects.⁵² The chosen cavity length is sufficient to enable the wavepackets of the turbulent spot to interact with the free shear layer of the cavity, while it is sufficiently short to avoid bypass transition affected by interaction of the cavity shear layer with the cavity downstream edge. The cavity length in the simulation corresponds to the shortest cavity length considered in the experiments. Dynamic similarity between the simulation and the experiments is realized by matching the flow Reynolds number to within 15%.

For all three simulations, the static pressure at the outflow boundary is kept fixed in an area-averaged sense. The inflow data for Simulation 2 and 3 are mapped from a plane in Simulation 1 that is $28 δ_{0}^{*}$ upstream of the domain outflow boundary (Fig. 4). This distance to the outflow boundary was established by Brinkerhoff and Yaras¹ to be sufficient to avoid any artificial effects of standing waves that may result due to the pressure-field constraint imposed at this boundary.

2. Base flow and turbulent spot initialization

Prior to introducing the spot, a laminar boundary layer is developed on the no-slip surfaces of Simulations 1, 2, and 3, including a statistically steady cavity flow in Simulation 3. A turbulent spot is then artificially triggered by impulsively introducing a spatially uniform jet of air at the inlet of the jet pipe with a velocity $7.7 U_{0}$ ⁠, for a duration of $Δ τ_{jet} = 27$ time units, where $τ = t U_{0} / δ_{0}^{*}$ is a dimensionless time coordinate and t is dimensional time. The jet is activated at τ = 0 into a developed undisturbed laminar boundary layer, and Simulation 1 is continued until the laminar boundary layer fully recovers at τ = 330. The instantaneous velocity data on the mapping plane in Simulation 1 is mapped to both Simulation 2 and 3 over a $Δ τ = 200$ period (⁠ $130 \leq τ \leq 330$ ⁠), during which the spot and its wake region are fully convected through the mapping plane. The last time step for which the wake region is observable on the mapping plane occurs at τ = 319. The streamwise extent of the wake region is defined by the most upstream coordinate where the streamwise component of perturbation velocity is 5% of the freestream velocity. A time-independent velocity field corresponding to the local Blasius profile is imposed at the inflow of Simulations 2 and 3 for $τ > 330$ ⁠. Temporal integration is continued until τ = 595 and τ = 905 for Simulations 2 and 3, respectively.

3. Spatial grid

The spatial grids consist of hexahedral cells mapped in a structured layout. To resolve all aspects of flow turbulence, the spatial and temporal resolutions must be on the order of the Kolmogorov length and time scales. The Kolmogorov length scale typically has a minimum value of $η^{+} \approx 2$ close to the no-slip surface in turbulent boundary layers.^53,54 The superscript + denotes normalization by viscous length and time scales, which for the present study are computed based on spatially averaged friction velocity, $u_{τ} = \sqrt{τ_{w} / ρ}$ ⁠, under the footprint of the spot at its most mature state in the absence of the cavity. The grid-node distributions are summarized in Table II and Fig. 5. The nodes are clustered in regions where relatively high gradients in flow variables are anticipated. In regions of nonuniform node-spacing, the rate of change of node spacing is kept below 10% in all three grid directions. The smallest grid-node spacing in the x, y, and z directions outside the jet pipe corresponds to about four to ten times the Kolmogorov scale. This has been shown to be suitable for accurate direct numerical simulations of turbulent boundary layers.^55–57 The jet subdomain is discretized with 60 nodes in each of the mutually perpendicular directions normal to the jet flow (x and z), with uniform values of $Δ x^{+} = 0.52$ and $Δ z^{+} = 0.52$ ⁠. The direction parallel to the jet flow is discretized with 88 nodes distributed as shown in Fig. 5(c), where the coordinates are defined such that ξ = 0 coincides with the outlet of the jet pipe and increases toward its inlet.

TABLE II.

Spatial grid specifications.

	Node spacings			Node counts
	$Δ x^{+}$	$Δ z^{+}$	$Δ y_{wall}^{+}$	N_x	N_y	N_z
Simulation 1	20	8	0.7	345	82	197
Simulation 2	20	7	0.7	456	82	266
Simulation 3	See Fig. 5(a)	8^a	0.7	707	82 (⁠ $y^{+} \geq 0$ ⁠); 95 (⁠ $y^{+} < 0$ ⁠)	437

	Node spacings			Node counts
	$Δ x^{+}$	$Δ z^{+}$	$Δ y_{wall}^{+}$	N_x	N_y	N_z
Simulation 1	20	8	0.7	345	82	197
Simulation 2	20	7	0.7	456	82	266
Simulation 3	See Fig. 5(a)	8^a	0.7	707	82 (⁠ $y^{+} \geq 0$ ⁠); 95 (⁠ $y^{+} < 0$ ⁠)	437

^a

Between $- 83 \leq z / δ_{0}^{*} \leq 83$ ⁠. Otherwise increases by 10% per node up to the spanwise boundaries.

FIG. 5.

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(a) Streamwise and (b) wall-normal node spacing for Simulation 3. (c) Node spacing in the jet pipe in the direction parallel to the jet flow, where ξ = 0 is located at the jet pipe outlet.

4. Solution method

The incompressible form of the mass and momentum equations is solved with the commercial software package ANSYS CFX 16.2. Spatial discretization is based on the vertex-centered finite volume approach with values of velocity and pressure collocated at grid nodes. Interpolation of nodal values to integration points on the control-volume surface is performed in manners equivalent to second-order central differencing for the diffusion terms, and a blend of first-order upwind and second-order central differencing schemes for the convection terms. Temporal derivatives are discretized with second-order Euler backward differencing. For Simulation 1, a time step of $Δ t = 7 \times 10^{- 6}$ s is chosen such that the pulsed jet takes ten timesteps to penetrate a distance of $δ_{0}^{*}$ into the undisturbed boundary layer. For Simulations 2 and 3, a time step of $Δ t = 3.5 \times 10^{- 5} s$ (⁠ $Δ t^{+} = 0.11$ ⁠) is chosen, where the + superscript denotes normalization by the viscous time scale, $ν / u_{τ^{2}}$ ⁠. This is a more conservative temporal resolution than the recommended value of $Δ t^{+} = 0.2$ to resolve the dynamics of coherent structures of boundary layer turbulence.⁵⁸

The discretized mass and momentum equations are solved in each time step by computing up to eight iterations that converge the coefficients of the linearized equations, with the solution of the linearized equations in each iteration realized through a W-type algebraic multigrid cycle over six grid levels. At each grid level, one and three solution sweeps are, respectively, performed during the restriction and prolongation passes of the cycle. For each time step, the solution is declared as converged when the root mean square normalized residual values of the governing equations are reduced to less than $10^{- 6}$ ⁠. Simulation 1 and Simulation 2 were partitioned and solved by Brinkerhoff and Yaras¹ with 20 Intel L5410 Xeon processors. Simulation 3 is partitioned and solved with 128 AMD Opteron 6272 processors.

This algorithm has been demonstrated by Valtchanov, Brinkerhoff, and Yaras⁵⁹ and earlier studies by the authors' research group to be suitable for simulating boundary layers in transitional and turbulent regimes through a benchmark direct numerical simulation of zero-pressure-gradient turbulent boundary layer (ZPGTBL). The Reynolds number of the ZPGTBL simulation is ${Re}_{θ} = 900$ based on the boundary layer momentum thickness, which is higher than the present simulation's ${Re}_{θ} = 401$ ⁠, calculated based on the average momentum thickness in the spot at its most mature state. As such, the ZPGTBL study realizes a broader spectrum of turbulence scales and thus serves as a more stringent validation. The grid nodes of the ZPGTBL are spaced at $Δ x^{+} = Δ z^{+} = 3$ ⁠. The y⁺ value of the first node from the no-slip boundary is 0.85 and the node spacing closely follows the distribution shown in Fig. 5(b). The result of the validation study, shown in Fig. 6, achieves excellent agreement with the turbulence-kinetic-energy budgets and streamwise velocity fluctuation profiles of Jimenez et al.⁶⁰ for a Reynolds number of ${Re}_{θ} = 1100$ ⁠. Further validation of the present numerical algorithm by the authors' research group for boundary and free shear layer flows in both compressible and incompressible flow conditions,^55–57,61 showing good agreement in velocity fluctuation spectra, turbulence-kinetic-energy production rate budgets, and mean velocity profiles, complement the results observed in the validation study of Valtchanov, Brinkerhoff, and Yaras.⁵⁹ The same algorithm was used by Brinkerhoff and Yaras¹ for the study of turbulent spots. In that study, it was demonstrated that owing to the relatively large coherent structures prevailing in a turbulent spot, these structures can be suitably resolved with somewhat coarser spatial grids based on $Δ x^{+} = 22, Δ z^{+} = 11$ ⁠, and $Δ y_{wall}^{+} = 0.85$ ⁠. The grid resolution of the present study is based on these findings.

FIG. 6.

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Comparison of the direct numerical simulations of Valtchanov, Brinkerhoff and Yaras⁵⁹ and Jimenez et al.:⁶⁰ (a) rates of turbulence-kinetic-energy production, $P^{+} = - {\bar{u_{i} u_{j}}}^{+} \partial {\bar{U}}^{+} / \partial x_{j}^{+}$ ⁠, and dissipation, $ε^{+} = {\bar{\frac{\partial u_{i}}{\partial x_{j}} \frac{\partial u_{i}}{\partial x_{j}}}}^{+}$ and (b) profiles of streamwise velocity fluctuation.

B. Inviscid modeling in the Lagrangian reference frame

Hairpin vortices of turbulence induce velocity on themselves and other hairpin vortices in their vicinity primarily through inviscid mechanisms. To help shed light on the flow physics, the computational approach described in Sec. III A is complemented by a localized Lagrangian vortex-filament-based inviscid model. This model is initialized with a single Kelvin–Helmholtz roller in the form of a vortex filament embedded in an idealized background two-dimensional streamwise velocity distribution based on the time-averaged velocity over the cavity obtained from the Eulerian model described in Sec. III A. This velocity distribution is then perturbed with a spanwise-periodic sinusoidal perturbation in the x and y components of velocity (⁠ $u'$ and $v'$ ⁠) with an amplitude of about 10% of freestream velocity and a wavelength of about 100 viscous wall units (⁠ $4.5 δ_{0}^{*}$ ⁠). This computational model is summarized in Fig. 7. The velocity at a point of interest, A, is the vector sum of the background velocity, ${\vec{V}}_{background}$ ⁠, and the velocity induced by the vortex filament as described by the Biot–Savart law,

\vec{V} = {\vec{V}}_{background} + \frac{Γ}{4 π} \int_{- \infty}^{\infty} \frac{{\vec{r}}_{m} \times d {\vec{s}}_{m}}{| {\vec{r}}_{m} |^{3}},

(3)

\vec{V} \approx {\vec{V}}_{background} + \frac{Γ}{4 π} \sum_{m} \frac{{\vec{r}}_{m} \times Δ {\vec{s}}_{m}}{| {\vec{r}}_{m} |^{3}},

(4)

where Γ denotes the circulation of the vortex filament set to a value of $δ_{0}^{*} U_{0}$ based on the roller circulation observed in the Eulerian model in the absence of the turbulent spot. The remaining terminology is summarized in Fig. 7.

FIG. 7.

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Schematic of the vortex filament computational model: (a) vortex filament observed at a mature stage of development; (b) a magnified view of segment $Δ s_{m}$ identified in (a).

The modeled vortex filament is $40 δ_{0}^{*}$ long. This is close to ten times the spanwise wavelength of the dominant instability mode expected in turbulent spots⁶² (⁠ $Δ z = 100^{+} = 4.5 δ_{0}^{*}$ ⁠). Discretization of the vortex filament into 200 segments, each with an initial length of $0.2 δ_{0}^{*}$ ⁠, produces results that are insensitive to the segment length. During the evolution of the vortex filament, the segments are subdivided whenever they exceed a length of $0.75 δ_{0}^{*}$ due to stretching, thus ensuring consistency of spatial resolution over the simulated development time. A time-independent finite radius of $r_{c} = 0.6 δ_{0}^{*}$ is assigned to each segment to prevent numerical instability, where the segment is not able to induce a velocity on itself within the core volume defined by this radius. The inviscid nature of this Lagrangian model is justified on the basis of the Kelvin–Helmholtz roller taking less than 15% of the viscous diffusion time scale, $r_{c}^{2} / ν$ ⁠, to convect across the length of the cavity. The vortex filament is developed in time using the same time step size of the direct numerical simulation described in Sec. III A. The vortex filament takes 200 timesteps to convect from its initial position set at the cavity surface halfway along the length of the cavity to the downstream edge of the cavity.

IV. FREESTREAM OPERATING CONDITIONS, BOUNDARY LAYER DEVELOPMENT, AND CAVITY FLOW DEVELOPMENT WITHOUT A TURBULENT SPOT

In the experiment, the location of the cavity upstream edge corresponds to a Reynolds number of $195 000 \pm 5000$ based on the freestream velocity and the distance from the test surface leading edge. The freestream velocity of $U_{0} = 6 m / s$ and turbulence intensity of 0.4% is uniform throughout the length of the test section. Upstream of the cavity, the undisturbed Blasius boundary layer with a shape factor that remains in the 2.4 to 2.8 range grows to a displacement thickness of 1.24 mm at the streamwise location of the spot-generating jet (⁠ $δ_{0}^{*}$ ⁠) and a momentum thickness of 0.9 mm at the upstream edge of the cavity (θ_s). $δ_{0}^{*}$ and θ_s are used as reference length scales in the remainder of the paper. Downstream of the shortest cavity considered (⁠ $L / θ_{s} = 46.5$ ⁠), the boundary layer remains laminar at least up to the most downstream measurement location, which is positioned at 1000 mm (⁠ $806 δ_{0}^{*}$ ⁠) from the test surface leading edge. Immediately downstream of the cavity, the time-averaged velocity profile of the boundary layer is relatively less full corresponding to a shape factor of 1.9, which gradually returns to a value of 2.6 at 3.5 cavity lengths (⁠ $120 δ_{0}^{*}$ ⁠) from the cavity. Two-dimensionality of the cavity flow for all three cavity lengths was confirmed through measurements of the laminar cavity velocity field at several spanwise locations.

Consistent with the experimental observations, the boundary layer in the simulations remains laminar with a shape factor of 2.5 up to the cavity, with a displacement thickness that agrees with the corresponding Blasius value to within 4%. The simulation flow Reynolds number based on the streamwise location of the cavity and the freestream velocity is 170 600, and the freestream turbulence intensity is 0.2%. The low level of freestream turbulence is the result of temporal velocity fluctuations caused by round-off errors in the computation and was not set explicitly. The time-averaged states of the cavity shear layer and the boundary layer downstream of the cavity are computed from seven full vorticity shedding cycles once the computed flow attains a statistically steady state. The cycle-to-cycle variation in streamwise peak velocity oscillation amplitude was noted to remain within 6%. Comparable with the experimental observations, the shape factor of the boundary layer varies from 2.1 immediately downstream of the cavity to 2.5 at three cavity lengths downstream. The reduction in the shape factor value to 2.1 is accompanied by a 12% reduction in the boundary layer displacement thickness with reference to the corresponding Blasius value. At the downstream location where the shape factor returns to the Blasius value of 2.5, the displacement thickness is only 3% below the corresponding Blasius value. These observations indicate that the boundary layer is able to fully recover from its encounter with the cavity in this particular configuration.

In the simulation with $L / θ_{s} = 46.5$ cavity length, feedback-driven oscillations of $0.2 U_{0}$ magnitude are observed in the streamwise velocity of the shear layer over and downstream of the cavity at a frequency of ${St}_{θ_{s}} = 0.015$ consistent with published literature (Fig. 2). The length of the cavity corresponding to $L / θ_{s} = 46.5$ is very close to the threshold separating the oscillating and nonoscillating flow regimes.^27,32 This threshold has been shown to be sensitive to Reynolds number and cavity depth,^33–35 resulting in some uncertainty regarding the existence of oscillations in close proximity of the threshold. The absence of flow oscillations at the $L / θ_{s} = 46.5$ value in the present experiments may be due to this variability. To nonetheless allow for a complementary analysis of the computed and experimental results, two additional cavity lengths corresponding to $L / θ_{s} = 54.4$ and $L / θ_{s} = 70.0$ were included in the experimental study. Both of these cavity configurations produce the expected feedback-driven oscillatory flow field, with Strouhal number values of ${St}_{θ_{s}} = 0.0095$ and 0.0084, respectively. This trend in oscillation frequency with cavity length is consistent with the literature (Fig. 2). The amplitude of velocity oscillation increases from $0.05 U_{0}$ to $0.15 U_{0}$ as $L / θ_{s}$ is increased from 54.4 to 70.0, which is also in agreement with past studies.^43,63

The boundary layer downstream of the cavity remains laminar for $L / θ_{s} = 46.5$ in both the simulation and the experiments, but transitions to turbulence in the experiments with $L / θ_{s} = 54.4$ and 70.0. The transition to turbulence in these two cases is observed to be completed by six cavity lengths from the downstream edge of the cavity. This location is determined based on the turbulence intermittency in the boundary layer reaching a value of 100%, with the intermittency quantified from time traces of the local velocity using the method of Volino et al.⁶⁴

The time-averaged velocity field at the streamwise midpoint of the cavity is plotted in Fig. 8. The agreement between experimental and computational results is noted to be favorable. The time-averaged shear layer over the cavity is further characterized through streamwise variation of vorticity thickness determined with this velocity field as plotted in Fig. 9. This thickness is defined as

δ_{ω z} = U_{0} / {(d \bar{u} / d y)}_{max} .

(5)

In the figure, $δ_{ω z}$ is normalized by the same at the cavity upstream edge. Vorticity thickness is preferred for this characterization over alternative shear layer thicknesses, such as displacement and momentum thicknesses, as the latter thicknesses are found to be less informative on the flow physics in the presence of recirculation within the cavity. The streamwise variation of $δ_{ω z}$ in the present study is noted to compare favorably with that of Sarohia.²⁷ This is not surprising given the similar values of L/D and $L / θ_{s}$ for the present study and the study of Sarohia.²⁷ The agreement with the data of Basley et al.³⁷ is also reasonable. However, the data of Chang et al.³⁹ suggest a notably higher rate of shear layer growth, particularly over the second half of the cavity length. This may be explained by the formation of hairpin vortices at $x / L = 0.5$ that the authors attribute to three-dimensional instability originating in their cavity. This is not observed in the present study, likely due to the larger cavity depth.

FIG. 8.

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Wall-normal profile of time-averaged streamwise velocity at the streamwise midpoint location in the cavity (⁠ $L / θ_{s} = 46.5$ ⁠).

FIG. 9.

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Streamwise development of vorticity thickness over the cavity.

Figure 10 shows the instantaneous distribution of the spanwise component of perturbation vorticity on the midspan plane at a statistically steady state in the simulation. The perturbation vorticity vector, $\vec{ω'}$ ⁠, is defined as the curl of the perturbation velocity vector, $\vec{u'}$ ⁠. $\vec{u'}$ is determined by subtracting the instantaneous velocity, $\vec{u}$ ⁠, from the time-averaged velocity, $\vec{\bar{u}}$ ⁠. Kelvin–Helmholtz rollers are observed to develop within the cavity shear layer and are identified with the black-colored contour lines of $ω_{z}^{'} δ_{0}^{*} / U_{0} = - 0.0158$ ⁠. The rollers induce downwash on their downstream side and upwash on their upstream side, creating time-periodic positive and negative perturbation in the local velocity field. The rollers are further characterized by their circulation, $Γ'$ ⁠. In each instance, this circulation is determined by summing up $ω_{z}^{'}$ over the area enclosed by the $ω_{z} δ_{0}^{*} / U_{0} = - 0.0158$ contour. The circulation values noted in the figure are obtained by averaging the values over six shedding cycles. The rollers reach their peak circulation immediately prior to the impingement on the cavity downstream edge. During the impingement, a smaller portion of each roller is swept into the cavity while the majority is convected downstream. Downstream of the cavity, the interaction of the rollers with the wall results in the generation of perturbation vorticity of the opposite sign beneath each roller, causing the rollers to weaken with streamwise distance. The simulation domain is not long enough for the vorticity to completely dissipate as the counter-rotating vorticity beneath the rollers also weakens with streamwise distance. This computational result is in agreement with the experimental results, where velocity oscillations remain detectable 15 cavity lengths downstream of the cavity, which is near the outlet of the test section. In close vicinity of the cavity downstream edge, the rollers have an average streamwise spacing and convection speed of $23 θ_{s} \pm 1 θ_{s}$ and $0.36 U_{0} \pm 0.01 U_{0}$ ⁠, respectively. The convection speed agrees well with the speed of $0.4 U_{0}$ reported by Chang et al.³⁹ The noted streamwise spacing and convection velocity corresponds to a Strouhal number value of $S t_{θ_{s}} = 0.015$ ⁠. This value is consistent with the Strouhal number calculated from the measured time trace of velocity in the cavity shear layer, $0.1 L$ upstream of the downstream edge of the cavity.

FIG. 10.

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Instantaneous spatial distribution of $ω_{z}^{'} δ_{0}^{*} / U_{0}$ at $z = 0$ ⁠. Negative vorticity in this view corresponds to a clockwise sense of rotation.

V. RESULTS AND DISCUSSION

A. Development of the turbulent spot upstream of the cavity

The turbulent spot measured in the experiment at distances of $175 θ_{s}$ (⁠ $x / δ_{0}^{*} = 121.0$ ⁠) and $8 θ_{s}$ (⁠ $x / δ_{0}^{*} = 241.9$ ⁠) upstream of the $46.5 θ_{s}$ -long cavity is visualized through the ensemble-averaged perturbation velocity distribution in Fig. 11. As noted previously, $δ_{0}^{*}$ denotes the boundary layer displacement thickness at the jet coordinate. The y-τ plane shows the turbulent spot centerline while the y-z and z-τ planes are located such that the dominant features of the turbulent spot are most clearly visualized. The y-τ plane shows a turbulent spot of conventional shape, with a height, which is about twice the local boundary layer thickness. A larger region of negative perturbation velocity resides over a smaller region of positive perturbation velocity. An overhang region is observed downstream of the turbulent spot while a wake with a positive perturbation velocity that has a stabilizing (calming) effect is observed on its upstream side. These observations are consistent with experimental and numerical studies on turbulent spots subject to zero streamwise pressure gradient.^1,20 In the z-τ plane, the turbulent spot follows a conventional shape approximating a downstream-pointing arrowhead. High- and low-speed streaks are visible in both the z-τ and y-z planes. The spanwise streak spacing is determined to be $λ_{z}^{+} \approx 160$ ⁠. This value is comparable to those reported in published literature.^4,10 The spanwise-smeared appearance of the streaks when observed through ensemble-averaged quantities is likely due to spanwise meandering of the streaks, a phenomenon commonly observed unless diminished by high streamwise acceleration driven by strong favorable streamwise pressure gradients.^20,49 The perturbation velocity magnitude for the high-speed streaks near the wall reaches 30% of the freestream velocity. This is just above the threshold of 26% suggested to be necessary for the sinuous streak instability to be significant in a zero-pressure-gradient boundary layer.¹¹ This may further contribute to the spanwise smearing of the streaks.

FIG. 11.

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Perturbation velocity in the y-τ, y-z, and z-τ planes at the experimental measurement planes (a) $x / δ_{0}^{*} = 121.0$ and (b) $x / δ_{0}^{*} = 241.9$ ⁠. The y-τ plane is located at the spot centerline. Dashed line shows the location of the y-z plane. Dotted line shows the location of the z-τ plane. Black solid lines approximate the perimeter of the spot with $u_{rms} / U_{0} = 0.06$ ⁠. Plots are not to scale.

The edges of the spot are located using the method outlined in Sec. II. Using data obtained at streamwise locations of $x / δ_{0}^{*} = 121.0$ ⁠, 181.5 and 241.9, the lateral growth of the spot as it convects in the streamwise direction is determined to correspond to a spreading angle of 7.3°. This is consistent with values quoted in the literature for zero-pressure-gradient boundary layers.^6,16 The leading and trailing edge celebrities are calculated to be $0.6 U_{0}$ and $0.3 U_{0}$ ⁠, respectively, which are lower than the often-cited values of $0.9 U_{0}$ and $0.55 U_{0}$ ⁠.^17,20 This difference may be explained by the lower wall-normal coordinates at which the leading and trailing edges of the relatively young spot are situated. The rate of streamwise growth of the turbulent spot, as dictated by the difference between the leading- and trailing edge velocities, remains consistent with the literature.

Large-scale turbulent spot features such as streak spacing, streak perturbation velocity magnitude, and spot longitudinal and lateral spreading rates are consistent between the experiment and the simulations. Further details on the internal structure of the spot are more readily gleaned from the simulations. In the absence of cavity effects (Simulation 1 and 2), these details were described comprehensively by Brinkerhoff and Yaras.¹ To serve as a reference for the structure of the spot past the cavity, a summary of the observations of Brinkerhoff and Yaras¹ is presented here, with an emphasis on flow structures that will be shown to play an important role in the spot/cavity flow interaction.

Figure 12 shows the turbulent spot as it develops from a young spot with only two generations of hairpin vortices [Fig. 12(a)] until it reaches $30 δ_{0}^{*}$ upstream of the cavity upstream edge [Fig. 12(b)]. The spot is visualized with isosurfaces of the second invariant of the velocity gradient tensor, denoted Q, normalized by ${(U_{0} / δ_{0}^{*})}^{2}$ ⁠. Q is defined as

Q = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{i}} \frac{\partial u_{j}}{\partial x_{j}} - \frac{\partial u_{i}}{\partial x_{j}} \frac{\partial u_{j}}{\partial x_{i}}),

(6)

where Einstein summation is implied over the indexed terms. The illustrated isosurface value is $Q / {(U / δ_{0}^{*})}^{2} = 2 \times 10^{- 4}$ ⁠, which allows the dominant vortical structures to be visible most clearly. Upon the introduction of a wall-normal jet pulse into the laminar boundary layer at τ = 0, a pair of counter-rotating, streamwise-oriented vortices (labeled A) form downstream of the jet. The induced velocity field of the A vortices reorients the local background spanwise vorticity to develop a wall-normal component. The reoriented vorticity is then stretched by the background shear in the x-y plane and forms the initial wavepackets (labeled B and C) that straddle the A vortices. By $τ = 118$ ⁠, the turbulent spot contains about two generations of wavepackets, with the more mature generation (B and C) evolving into the familiar hairpin shape. By $τ = 301$ ⁠, the turbulent spot contains several generations of wavepackets, most of which are situated at a wall-normal position comparable to the centerline position of the Kelvin–Helmholtz rollers over the cavity. A few of the larger vortical structures in the leading edge region of the spot, such as C, are located at $y / δ_{0}^{*} \approx 10$ ⁠, which is about twice the wall-normal distance of the rollers' uppermost portions over and downstream of the cavity (Fig. 10). At this moment, the turbulent spot has developed a pattern of growth that will be present for its entire lifetime if it is allowed to convect over the surface without any geometric or other external perturbations: (1) older vortical structures, such as C, D, and G, are enlarged and concentrated near the leading edge of the spot; (2) hairpin vortices of a range of spatial scales occupy the interior of the spot, where turbulence is sustained by the perpetual generation of new smaller hairpin vortices in streamwise-aligned groups (wavepackets) via the induced and destabilizing effects of the existing larger hairpin vortices; and (3) the laminar flow along the side edges of the spot is similarly destabilized to produce wavepackets in this previously undisturbed region, thus promoting lateral spot growth.¹ Mutual interactions of the hairpin vortices within a mature spot can cause the shape of the vortices to deviate significantly from the idealized hairpin shape. For example, much of the streamwise-oriented structures noted in Fig. 12(b), such as structure G, evolved originally from hairpin vortices.

FIG. 12.

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The turbulent spot upstream of the cavity visualized through isosurfaces of $Q / {(U / δ_{0}^{*})}^{2} = 2 \times 10^{- 4}$ at (a) τ = 118 and (b) $τ = 301$ ⁠.

The leading edge of the spot arrives at the cavity upstream edge at approximately τ = 330. Thus, the spot structure shown in Fig. 12(b) at τ = 301 closely represents the spot at the start of the interaction with the cavity.

B. Development of the turbulent spot over and downstream of the cavity

The first measurement plane downstream of the cavity is at $4 δ_{0}^{*}$ distance from the downstream edge of the cavity. The turbulent spot measured at this location (⁠ $x / δ_{0}^{*} = 287.3$ ⁠) is illustrated in Fig. 13(a). In contrast to the spot structure upstream of the cavity illustrated in Fig. 11(b), the high-speed streaks positioned in close vicinity of the wall are notably weaker, disappearing completely in the middle portion of the spot's length. Measurements over the cavity, at $22 δ_{0}^{*}$ (65% of cavity length) from the cavity upstream edge, indicate that velocity fluctuations due to the spot turbulence extend into the cavity by about 2.0 to $2.5 δ_{0}^{*}$ ⁠. This suggests the spot to slightly descend into the cavity as it travels over the cavity. This is consistent with the expected mutually-induced downward motion of the streamwise legs of the hairpin vortices in the absence of a solid boundary that would prevent such a motion. As a result of this downward motion, the high-speed streaks created by the induced motions of the hairpin vortices appear to impinge on the downstream wall of the cavity and get at least partially swept into the cavity instead of convecting over the cavity downstream edge. By $126.6 δ_{0}^{*}$ distance from the cavity downstream edge (⁠ $x / δ_{0}^{*} = 408.4$ ⁠), shown in Fig. 13(b), the high-speed streaks are observed to regain their strength. However, both low- and high-speed streaks appear merged with their neighboring streaks compared to the observation upstream of the cavity. This suggests a less organized distribution of the streaks and/or increased meandering, and is consistent with past experimental and numerical studies of turbulent boundary layers downstream of a cavity.^39,42

FIG. 13.

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Perturbation velocity in the y-τ, y-z, and z-τ planes at the experimental measurement planes (a) $x / δ_{0}^{*} = 287.3$ and (b) $x / δ_{0}^{*} = 408.4$ ⁠. The y-τ plane is located at the spot centerline. Dashed line shows the location of the y-z plane. Dotted line shows the location of the z-τ plane. Black solid lines approximate the perimeter of the spot with $u_{rms} / U_{0} = 0.06$ ⁠. Plots are not to scale.

The streamwise variation of the spot half-width is plotted in Fig. 14. Upstream of the cavity, the half-spot spreading angle of 7.3° is constant over at least $120 δ_{0}^{*}$ streamwise distance, indicating that the mechanisms driving the spanwise growth of the spot in this region are well-established and independent of the disturbance that created the spot. The spreading angle downstream of the cavity is 16.1°, which is notably higher than the value of 7.3° measured upstream of the cavity. This indicates a significant increase in the lateral growth rate of the spot upon encountering the cavity. The relatively minor amount of deviation from a linear trend downstream of the cavity suggests that the physical mechanisms resulting in the increased lateral growth rate are established upstream of the cavity downstream edge and remain intact for at least $100 δ_{0}^{*}$ distance downstream. The mechanisms driving lateral growth will be discussed further in Sec. V B 4. Encountering the cavity also affects the streamwise growth rate of the spot, which will be discussed in Sec. V C.

FIG. 14.

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Variation of spot half-width, $s_{h w}$ ⁠, with streamwise location of the spot.

1. Development of the leading edge region of the spot downstream of the cavity

Prior to the arrival of the turbulent spot, Kelvin–Helmholtz (K–H) rollers over and downstream of the cavity are coherent and spanwise uniform. The leading edge region of the spot convects faster than the rollers, allowing it to overtake and interact with multiple rollers. Since most vortical structures within the leading edge region of the spot are larger-scale structures located high above the rollers, changes to these vortical structures and the K–H rollers are not observed until they reach a location, which is one to two-cavity lengths downstream of the cavity. By τ = 451, the spot leading edge has overtaken two K–H rollers and has traversed two-cavity lengths on the downstream surface. The spot at this instant in time is illustrated via isosurfaces of Q in Fig. 15(b). For reference, the spot at the same instant in time in the absence of the cavity is shown in Fig. 15(a). Vortical structure C, which is higher up than the other structures as per the color scale is essentially the same in Figs. 15(a) and 15(b), suggesting that it has avoided any interaction with the rollers. Streamwise-elongated vortical structure labeled G extends by $60 δ_{0}^{*}$ in the streamwise direction in the absence of the cavity [Fig. 15(a)]. In the presence of the cavity [Fig. 15(b)], this structure has been tilted in the direction of the rollers' induced velocity. The part of the G structure immediately downstream of roller 2 is convected toward the wall to a wall-normal coordinate, which is half of where this structure resides at in the absence of the cavity. Although such tilting of streamwise-aligned vortices is observed throughout the leading edge region of the spot, its impact on the composition of the spot is relatively limited as the tilted structures appear to evolve in a similar manner to the corresponding structures in the absence of the cavity. A notably larger effect of the cavity on the spot structure is observed near the coordinates of roller 2 (⁠ $310 \leq x / δ_{0}^{*} \leq 330$ ⁠) where a spanwise row of hairpin vortices not observed in the absence of the cavity, labeled H, is observed in place of roller 2. The mechanism of their formation is demonstrated by correlating their positions with the positions of high-speed streaks, shown in Fig. 16. In this figure, streamwise perturbation velocity is shown on a surface at $y / δ_{0}^{*} = 3.75$ ⁠, which intersects with the centerline of roller 2. High-speed streaks labeled J are present within the spot both with and without the cavity and have a spanwise spacing of approximately $λ_{z} = 12 δ_{0}^{*}$ ⁠. The locations of the H vortices coincide with the locations of these streaks. Examination of a time series of Q isosurfaces and perturbation velocity contours shows the following sequence of events: (1) as the high-speed streak overtakes roller 2, the initially spanwise vorticity of roller 2 reorients into the streamwise direction in accordance with the spanwise wavelength of the streaks; and (2) self-induced lift-up and the subsequent stretching into the streamwise direction causes the H vortices to form. The same process is observed for all rollers that are initially spanwise coherent and uniform that are subsequently overtaken by the spot. These hairpin vortices thus have a streamwise spacing identical to that of the K–H wavelength and a spanwise spacing matching the prevailing streak spacing of the spot.

FIG. 15.

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The turbulent spot visualized through isosurfaces of $Q / {(U / δ_{0}^{*})}^{2} = 2 \times 10^{- 4}$ at τ = 451 (a) in the absence, and (b) in the presence of the cavity.

FIG. 16.

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Magnified view of the H vortices shown in Fig. 15 overlaid with color contours of perturbation streamwise velocity on a surface at $y / δ_{0}^{*} = 3.75$ (a) in the absence, and (b) in the presence of the cavity.

2. Development of spot turbulence over and downstream of the cavity

By τ = 484, about half of the streamwise length of the turbulent spot has convected past the cavity. Figure 17(a) illustrates this moment from a top-down view with Q isosurfaces. For reference, Fig. 17(b) shows the spot at the same instant in time in the absence of the cavity. Qualitatively, larger-scale turbulent spot structures located relatively far from the no-slip surface appear unaltered by the encounter with the cavity [Fig. 17(a) vs Fig. 17(b)]. Downstream of the cavity downstream edge and close to the surface, the number of vortical structures appears to be notably larger in the presence of the cavity. When viewed from the bottom [Fig. 17(c)], most of these structures are oriented to be within 15° of the streamwise direction, and in a few instances appear to form an upstream-pointing, Λ-like shape. The formation of these structures can be traced back to the inverted hairpin vortices that formed directly over the cavity. Inverted hairpin vortices have heads that are located upstream and below their streamwise-oriented legs, and are typically suppressed in boundary layer turbulence due to the friction exerted by the wall.^65,66 Their presence over the cavity [Fig. 17(c)] is not surprising as the no-slip condition of the wall is absent there. The origin of these inverted hairpin vortices is explored in more detail in Sec. V B 3. Their heads extend ∼ $2.5 δ_{0}^{*}$ into the cavity, which is consistent with the elevated $u_{rms}$ observed at the same depth into the cavity in the experiments. When the inverted hairpin vortices impinge against the downstream edge of the cavity, the heads are swept into the cavity while the approximately streamwise-aligned legs convect downstream of the cavity. Thus, the legs of the inverted hairpin vortices appear to account for the majority of the near-wall vortical structures downstream of the cavity downstream edge.

FIG. 17.

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Turbulent spot viewed at $τ = 484$ with black-color lines locating the cavity edges. [(a) and (b)]: top view of isosurfaces of $Q / {(U / δ_{0}^{*})}^{2} = 2 \times 10^{- 4}$ ⁠; [(c) and (d)]: bottom view of isosurfaces of $Q / {(U / δ_{0}^{*})}^{2} = 2 \times 10^{- 4}$ ⁠; [(e) and (f)]: flood plot of the wall-normal component of perturbation velocity with contours of x component of perturbation vorticity, $| ω_{x}^{'} | δ_{0}^{*} / U_{0}$ ⁠, corresponding to a value of +0.25 (red color) and −0.25 (blue color).

Additionally, wavepacket regeneration is also observed over the cavity. Figures 17(e) and 17(f) show the perturbation wall-normal velocity on a plane located at $y / δ_{0}^{*} = 0.5$ in the presence and in the absence of the cavity, respectively. Due to the no-penetration condition at the wall, perturbation wall-normal velocity is low at all coordinates except directly over the cavity. Wall-normal velocity induced by hairpin vortices prevailing in the spot grows to $\pm 0.2 U_{0}$ at a streamwise distance of around 30% of the cavity length downstream of the cavity upstream edge, where it coincides with a high level of streamwise vorticity as shown by the contour lines. Similar wall-normal perturbation velocity and streamwise vorticity distributions have been observed in several previous studies.^39,42 Three streamwise-aligned upright hairpin vortices are observed to form at approximately the same time over the cavity, enclosed in a dashed ellipse in Fig. 17(e). These coordinates are occupied by a streamwise-continuous high-speed streak in the absence of the cavity, which suggests that the three hairpin vortices formed over the cavity belong to the same wavepacket. Since the corresponding wavepacket is not observed in the absence of the cavity, its accelerated formation over the cavity is suggested to also contribute to the elevated number of vortical structures downstream of the cavity.

The streamwise lengths of high- and low-speed streaks downstream of the cavity are observed to be shorter than those of the streaks within the spot in the absence of the cavity. Such loss in streamwise coherence has been observed in past studies of turbulent boundary layers that convect over cavities.^39,42 In addition to the shorter lengths, streaks over and downstream of the cavity are observed to be streamwise-wavy with wavelengths that correlate with the spacing between sequential generations of inverted hairpin vortices, which are themselves correlated with each spanwise K–H roller that prevails outboard of the spot. The reduction in length and increase in streamwise-waviness of the streaks may explain the spanwise-smeared ensemble-averaged appearance of the spot measured downstream of the cavity in the experiments.

3. Lagrangian modeling of the K–H roller distortion by the disturbance of a turbulent spot

A Lagrangian model described in Sec. III B was utilized to better understand the interaction of the rollers that develop through the K–H instability of the cavity shear layer with the streaky perturbations in velocity associated with the passing of a turbulent spot. The initially straight, spanwise-aligned vortex filament model was subjected to three different perturbations in velocity in three separate computations. First, a spanwise-sinusoidal perturbation in the streamwise component of velocity is applied. Second, the same perturbation is applied to the wall-normal component of velocity. Finally, the two mentioned perturbations in the streamwise and wall-normal directions are linearly superimposed in phase. The wavelength and amplitude of the sinusoidal perturbations are $4.5 δ_{0}^{*}$ and $0.055 U_{0}$ ⁠, respectively. These perturbation characteristics were chosen to approximate the perturbation of the cavity shear layer by the passing of a turbulent spot, albeit at a lower amplitude. The perturbations in all three cases are applied for the entire duration of the Lagrangian computations, which lasts for $Δ τ = 22$ ⁠, which is the time required for a roller to convect from the middle of the cavity to the cavity downstream edge. Figure 18 shows the state of the vortex filament at the end of the three computations. In Fig. 18(a), the vortex filament is only subject to a perturbation in the streamwise velocity component. This causes the initially spanwise vorticity to be reoriented into the streamwise direction according to the wavelength of the imposed sinusoidal perturbation. Self-induced lift-up effects of the streamwise-aligned vorticity reorient the distorted segments of the vortex filament in the wall-normal direction where they are then stretched by the background shear. This results in hairpin-shaped loops in the vortex filament that have a streamwise length of about $4 δ_{0}^{*}$ and a wall-normal height of around $1 δ_{0}^{*}$ ⁠. In Fig. 18(b), the vortex filament is subject to a perturbation in the wall-normal velocity component. This immediately reorients the initially spanwise vorticity into the wall-normal direction, prompting stretching in the streamwise direction by the background shear. The resulting streamwise growth rate of the hairpin-shaped loops is much higher than the growth rate observed for the vortex filament subject to only streamwise velocity perturbations. In the last scenario, shown in Fig. 18(c), where both streamwise and wall-normal components of velocity are perturbed in phase, the resulting vortex filament is even longer in the streamwise direction and appears to take on a shape and dimension that are the linear superposition of the flow developments described in Figs. 18(a) and 18(b). This is expected given the linear nature of the governing equations for such flows that are driven by inviscid phenomena.

FIG. 18.

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The state of vortex filaments at the end of the simulation period $Δ τ = 22$ subject to (a) $u'$ ⁠, (b) $v'$ ⁠, and (c) in-phase $u' + v'$ perturbations.

Figure 19 shows the effect of perturbation velocity amplitude on the temporal growth rate of the vortex filament, where the temporal grow rate is calculated by dividing the streamwise length of the filament at the end of the simulation normalized with $δ_{0}^{*}$ by the dimensionless time duration. The growth rate is noted to scale linearly on the streamwise perturbation amplitude, whereas the effect of wall-normal perturbation on the growth rate decreases with increasing perturbation amplitudes. These results were noted to be insensitive to the chosen perturbation wavelength. The case with perturbations in both streamwise and wall-normal directions is deemed to most closely approximate the hairpin vortex dynamics of a turbulent spot as the spot traverses over the cavity. This is because both streamwise and wall-normal perturbations associated with the velocity streaks in the spot are free to develop in the absence of a wall.

FIG. 19.

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Streamwise growth rate of an initially straight spanwise vortex filament subject to perturbation velocity of varying amplitudes.

To compare the results of the Lagrangian computation with streamwise and wall-normal velocity perturbations to the Eulerian computation, Fig. 20 shows inverted hairpin vortices formed over the cavity for two sequential rollers in the Eulerian computation. The vortex filament and the inverted hairpin vortices [Fig. 18(c) vs Fig. 20(b)] are noted to have very similar shapes: each of them has a “head” located at a depth of $2 δ_{0}^{*}$ into the cavity and each has legs pointed into the downstream direction with a tilt away from the cavity opening at a comparable angle. The highlighted inverted hairpin vortices in region A develop over the same time interval as the Lagrangian vortex filament and both grow to streamwise length and wall-normal height values of about $10 δ_{0}^{*}$ and $4 δ_{0}^{*}$ ⁠, respectively, comparing favorably to the results of the Lagrangian computation. The favorable agreement suggests that the inverted hairpin vortices formed over the cavity are products of the deformed spanwise vorticity grouped together by the K–H instability over the cavity. Inverted hairpin vortices are observed for nearly all roller generations while the spot is over the cavity, with the exception of the first two roller generations that only interact with the larger-scale structures near the leading edge of the spot. This exception is expected, as these structures are located farther away from the cavity opening and thus do not induce large enough velocity perturbation amplitudes to cause the formation of such inverted hairpin vortices directly over the cavity.

FIG. 20.

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A typical inverted vortex pair formed within the cavity visualized through isosurfaces of $Q / {(U / δ_{0}^{*})}^{2} = 2 \times 10^{- 4}$ at τ = 451. (a) Bottom view; (b) side view of only region A; (c) bottom view of only region A.

4. Interaction of the turbulent spot with the cavity shear layer along the lateral boundaries of the spot

In the absence of the cavity, the creation of new hairpin vortices at the lateral boundaries occurs mostly in the trailing half of the spot, as the “parent” hairpin vortices prevailing in this region of the spot are located closer to the wall.¹ In the presence of the cavity, this process of hairpin vortex creation appears to accelerate as the trailing half of the spot approaches the cavity downstream edge. Figure 21 shows isosurfaces of Q for the—z half of the spot viewed from the bottom at three different instances in time. Figure 22 shows the perturbation streamwise velocity field at one of these times (τ = 572). At τ = 539 [Fig. 21(a)], the trailing edge of the spot is just about to reach the cavity upstream edge and an inverted hairpin vortex labeled P0 is situated at the lateral edge of the spot. This vortex P0 is formed from a deformed K–H roller with the mechanisms described in Sec. V B 3, and is just about to impinge upon the cavity downstream edge. Between $τ = 539$ and $τ = 572$ ⁠, P0 impinges against the cavity downstream edge such that its head is swept into the cavity while the legs are convected downstream. At $τ = 572$ [Fig. 21(b)], the spot is terminated at its spanwise edge by a streamwise-oriented leg of the P0 vortex, which rotates in the opposite direction compared to an upright hairpin vortex that would occupy this region in the absence of the cavity. This induces upwash outboard of the turbulent spot and causes a low-speed streak with a magnitude of $- 0.1 U_{0}$ to form (Fig. 22). The low-speed streak reorients the initially spanwise vorticity concentrated in the roller outboard of the spot backward into the streamwise direction, which causes the formation of another inverted hairpin vortex, P1. Self-induced lift-up effect causes the head of P1 to be convected toward the wall, where torque exerted by wall shear appears to reduce the vorticity magnitude over time [Fig. 21(c)]. The inner-facing leg of P1 never forms completely, likely due to its viscous interaction with P0, which has a larger vorticity magnitude of the opposite sense of rotation. The process is repeated between $τ = 572$ and $τ = 605$ ⁠, which results in the formation of P2.

FIG. 21.

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The turbulent spot viewed from the bottom visualized through isosurfaces of $Q / {(U / δ_{0}^{*})}^{2} = 2 \times 10^{- 4}$ at (a) τ = 539, (b) $τ = 572$ ⁠, and (c) τ = 605.

FIG. 22.

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Flood plot of streamwise perturbation velocity in a y–z plane at $x / δ_{0}^{*} = 320$ and τ = 572 with contours of $Q / {(U / δ_{0}^{*})}^{2} = 2 \times 10^{- 4}$ superimposed.

Multiple groups of such flow structures are observed on both lateral edges of the spot within $100 δ_{0}^{*}$ downstream distance of the cavity downstream edge, which is consistent with the experimental results showing an increased lateral growth rate in the same streamwise region. In the simulation results, the edges of the turbulent spot are determined using threshold values for the magnitude of the streamwise component of vorticity. This approach follows the work of Strand and Goldstein⁶ and Brinkerhoff and Yaras.¹ Specifically, eight uniformly spaced values of $| ω_{x} | δ_{0}^{*} / U_{0}$ are averaged in the range of 0.06 and 0.19 and used as the threshold value. The uncertainty of the edge positions, and thus the spot dimensions, are calculated as the 95% confidence interval of this average. The coordinates of structures such as P0, P1, and P2 are consistently detected to coincide with the lateral boundaries of the spot, which suggests that the accelerated lateral growth of the spot over and downstream of the cavity can be attributed to these structures. To allow for comparison of the lateral growth rate between the present and past numerical studies of turbulent spots in the absence of a cavity, the maximum spot half-width is plotted with respect to the streamwise location of the spot's trailing edge, x_te, in Fig. 23. Since this abscissa makes it unclear when the spot encounters the cavity, a red circle is drawn to mark the spot trailing edge coordinate at the instant when the spot leading edge reaches the cavity upstream edge. The results of Strand and Goldstein⁶ show a spot, which is consistently smaller than the present study. This discrepancy is likely due to the larger initial disturbance of the present study in the form of a strong wall-normal jet, whereas the spot of Strand and Goldstein⁶ is triggered by the temporary presence of a small roughness element. The strong jet of the present study causes a rapid initial growth phase that results in a larger starting width for the spot. Before the spot encounters the cavity (x_te < 175), the lateral growth rate of the turbulent spot agrees favorably with the growth rate in the absence of a cavity of both the present study and the study by Strand and Goldstein.⁶ The lateral growth rate of the spot remains unchanged as the leading half of the spot convects over the cavity (⁠ $175 \leq x_{t e} < 225$ ⁠) and approximately doubles when the trailing half of the spot reaches close to the cavity downstream edge (⁠ $x_{t e} \geq 225 δ_{0}^{*}$ ⁠). The latter trend is expected since the inverted hairpin vortices are predominantly generated in the trailing streamwise half of the spot. The increase in lateral growth rate and the width of the spot normalized by $δ_{0}^{*}$ agree favorably with the corresponding experimental data that was presented in Fig. 14.

FIG. 23.

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Variation of spot half-width with streamwise location of the trailing edge of the spot. The γ values denote half-spot spreading angles based on line fits to corresponding segments of data.

C. The wake region of the turbulent spot in the presence of feedback-driven oscillations

The wake region is a ubiquitous region upstream of the turbulent spot where a fuller streamwise velocity profile enhances stability.²² The extra momentum that yields the fuller velocity profile is accounted for by the presence of high-speed streaks within the wake that are non-turbulent.^1,22,67 The formation of these streaks is promoted by wall-normal momentum transfer induced by perturbation spanwise vorticity associated with the turbulent spot in close proximity of the wall.¹ Defining the upstream boundary of the wake to coincide with the location where perturbation streamwise velocity is 4% of U₀ and the downstream boundary to coincide with the spot trailing edge, the streamwise extent of the wake is 40% of the spot's length at the instant when the spot trailing edge is at the upstream end of the cavity. Figure 24 visualizes the interaction of the spot's wake with the cavity shear layer with Q isosurfaces at three equally spaced instances in time. At $τ = 705$ ⁠, 80% of the wake length has already convected past the cavity downstream edge. The labeled K–H roller in Fig. 24(a) is about to impinge on the downstream edge of the cavity. As this impingement process is about to occur, the upwash and downwash motions associated with this roller are observed to be spanwise nonuniform as illustrated through the perturbation velocity plots in Figs. 25(a) and 25(b). This nonuniformity is explained by the deformation of the K–H roller through its interaction with the high-speed streaks prevailing in the wake of the turbulent spot. These streaks are located within the dashed rectangles drawn on Fig. 25(a) and are relatively faint due to their relatively low magnitude. At $τ = 727$ ⁠, the region of upwash induced by the K–H roller coincides with the downstream edge of the cavity and transfers fluid out of the cavity at a velocity magnitude of around $0.15 U_{0}$ [Fig. 25(d)]. In response, to locally satisfy conservation of mass, near-wall fluid immediately downstream of the cavity is accelerated in the streamwise direction and momentarily increases the streamwise component of velocity at this region [Fig. 25(c)]. Since the upwash is spanwise nonuniform and is correlated with the existing streaks in the wake, the correlated streamwise acceleration follows the same pattern and increases the streak perturbation velocity amplitude to $0.25 U_{0}$ ⁠. Since these strengthened streaks are on the upstream side of the roller, they are preceded by a region of relatively low perturbation streamwise velocity due to the downwash of the roller on its downstream side. Between $τ = 727$ and $τ = 749$ ⁠, the newly formed high-speed streaks reorient the spanwise vorticity of the K–H roller into the streamwise direction, resulting in the formation of a spanwise row of hairpin vortices [Fig. 24(c)]. This process is repeated for every roller that forms within the wake of the spot, although the time and hence streamwise distance required for each roller to reorganize into hairpin vortices is progressively longer toward the upstream end of the spot's wake. Since the streak-velocity amplitude within the spot's wake decreases in the upstream direction, each subsequent generation of high-speed streaks generated at the cavity downstream edge is amplified to progressively lower amplitudes, which in turn causes lesser reorientation of the roller's spanwise vorticity. Downstream of the cavity, wavepacket regeneration is observed along the legs of some of these roller-originating hairpin vortices, which implies that the length of the turbulent spot has effectively increased over the cavity. The simulation is not long enough to determine how long after the passing of the turbulent spot the cavity shear layer returns to its undisturbed state.

FIG. 24.

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The evolution of a K–H roller as it interacts with the wake region, visualized through isosurfaces of $Q / {(U / δ_{0}^{*})}^{2} = 2 \times 10^{- 4}$ at (a) τ = 705, (b) τ = 727, and (c) τ = 749. Arrow points to the cavity downstream edge.

FIG. 25.

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Perturbation streamwise [(a), (c), and (e)] and wall-normal [(b), (d), and (f)] velocity component at $y / δ_{0}^{*} = 1$ ⁠. Dash rectangles follow the same region within the wake as it convects over the cavity downstream edge. Arrow points to the cavity downstream edge.

The stated observations in the numerical study are supported by the experimental measurements. Figure 26 shows the $u_{rms}$ distribution $5.6 δ_{0}^{*}$ distance downstream of the cavity during the passing of the turbulent spot for the three cavity lengths tested. The extent of the wake region in the time dimension is visualized with perturbation velocity contours of $u' / U_{0} = \pm 0.04$ ⁠. For the cavity length of $L / θ_{s} = 46.5$ [Fig. 26(a)], the wake region remains relatively unchanged from its appearance observed just upstream of the cavity [Fig. 11(b)]. This seems to suggest that no significant changes occurs in the wake as a result of the interaction of the spot and its wake with the cavity. Figures 26(b) and 26(c) show the spot structure after interaction with cavities of $L / θ_{s} = 54.4$ and $L / θ_{s} = 70.0$ length, respectively. In these cases, feedback-driven oscillation develops with amplitudes of $0.05 U_{0}$ and $0.15 U_{0}$ ⁠, respectively. The size of the region of elevated $u_{rms}$ and its magnitude are noted to positively correlate with increasing oscillation amplitude. Specifically, the peak $u_{rms}$ value within the wake of the spot that convects past the longest cavity (⁠ $L / θ_{s} = 70.0$ ⁠) reaches a magnitude comparable to the values observed within the spot itself. Furthermore, at least 50% of the wake length has elevated values of $u_{rms}$ ⁠. Although this cavity is longer than the one studied numerically, the feedback-driven velocity oscillation amplitude of the experiment is within 25% of that observed in the simulation. Thus, the elevated $u_{rms}$ values observed in Fig. 26(c) support the simulation results where hairpin vortices are formed in the spot's wake from the rollers following their impingement with the cavity downstream edge. Since significantly elevated $u_{rms}$ values in the spot's wake exist only when feedback-driven oscillations accompany the upwash and downwash motions induced by the rollers, the experiment supports the idea that the upwash induced by the rollers plays a crucial role in the genesis of hairpin vortices within the wake. Toward the upstream end of the wake, values of $u_{rms}$ return to the background level. This again is consistent with the simulation results, where the formation of hairpin vortices was described to require progressively longer periods of time toward the upstream end of the wake.

FIG. 26.

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Distribution of $u_{rms} / U_{0}$ $5.6 δ_{0}^{*}$ distance downstream of the cavity downstream edge along the spot centerline: (a) $L / θ_{s} = 46.7$ ⁠, (b) $L / θ_{s} = 54.4$ ⁠, and (c) $L / θ_{s} = 70.0$ ⁠. White solid lines: $u' / U_{0} = 0.04$ ⁠; white dashed lines: $u' / U_{0} = - 0.04$ ⁠.

VI. CONCLUSIONS

The evolution of coherent structures within an artificially triggered turbulent spot convecting past a deep cavity with a nominal aspect ratio of $L / D = 0.1$ is studied with hotwire measurements in a closed-circuit wind tunnel. The experimental results are complemented by dynamically similar direct numerical simulations and Lagrangian computations of an idealized vortex filament representing a K–H roller of the cavity shear layer subject to spanwise-sinusoidal velocity perturbations to mimic the effect of the turbulent spot on the roller. The experiments are conducted with a Reynolds number of 195 000 at the cavity upstream edge with three different cavity lengths of $L / θ_{s} = 46.5$ ⁠, 54.4, and 70.0. The simulations are performed with a Reynolds number of 170 610 and a single cavity length of $L / θ_{s} = 46.5$ ⁠.

The interaction of the turbulent spot with the cavity shear layer is observed to produce a notable increase in the number of coherent vortical structures associated with the spot. Most of these flow structures are inverted hairpin vortices that are created through interaction of the K–H rollers of the cavity shear layer with the trailing half of the spot. Along the lateral edges of the spot, the K–H rollers of the cavity shear layer are deformed to approximately the shape of an inverted hairpin vortex. In each instance, the induced velocity of this flow structure reorients the spanwise vorticity concentrated in the K–H roller on its outboard side into the streamwise direction, forming a new vortex that in turn repeats the process farther along the span of the roller. This process is noted to result in approximately doubling of the lateral growth rate of the spot, which continues to remain in effect as the spot convects over the surface downstream of the cavity. In the wake of the turbulent spot, the K–H rollers upstream of the turbulent spot are observed to deform into spanwise-wavy patterns by the streamwise velocity streaks in the wake of the spot. Interaction of these deformed K–H rollers with the downstream edge of the cavity is shown to strengthen the prevailing streamwise streaks, which in turn promote the formation of spanwise rows of hairpin vortices from the deformed K–H rollers. This development is noted to be correlated with the presence of strong feedback-driven oscillations in the cavity flow and results in a significant increase in the streamwise size of the turbulent spot.

ACKNOWLEDGMENTS

The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial support of the present study.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

NOMENCLATURE

D: cavity depth (m)
f: cavity feedback-driven oscillation frequency (Hz)
L: cavity length (m)
m: vortex segment index
n: number of spot generation and measurement cycles
N_x: number of nodes in the streamwise direction in the direct numerical simulation
N_y: number of nodes in the wall-normal direction in the direct numerical simulation
N_z: number of nodes in the spanwise direction in the direct numerical simulation
P⁺: nondimensionalized turbulence-kinetic-energy production rate
Q: second invariant of the velocity gradient tensor (⁠ ${1 / s}^{2}$ ⁠)
${\vec{r}}_{m}$: vector distance from vortex segment m (m)
r_c: minimum radius of induction by the vortex filament (m)
${St}_{θ_{s}}$: Strouhal number based on the momentum thickness at the cavity upstream edge, ${St}_{θ_{s}} = f θ_{s} / U_{0}$
${\vec{s}}_{m}$: vector length of vortex segment m
s_hw: turbulent spot half-width (mm)
t: time (s)
u: instantaneous streamwise velocity (m/s)
u_i, u_j: instantaneous velocity vector in index notation (m/s)
u_l: local undisturbed (laminar) streamwise velocity (m/s)
$u_{rms}$: root mean square of the velocity fluctuation (m/s)
$u_{τ}$: friction velocity (m/s)
$u'$: instantaneous perturbation streamwise velocity (m/s)
U₀: inlet and free stream streamwise velocity (m/s)
$\bar{u}$: time-averaged streamwise velocity in the numerical simulation (m/s)
$\tilde{u}$: ensemble-averaged streamwise velocity (m/s)
v: instantaneous wall-normal velocity (m/s)
$v'$: instantaneous perturbation wall-normal velocity (m/s)
$\vec{V}$: instantaneous velocity vector used with the vortex filament method (m/s)
${\vec{V}}_{background}$: background velocity vector used with the vortex filament method (m/s)
x_i, x_j: coordinate written in index notation (m)
x_te: streamwise coordinate of the turbulent spot trailing edge (m)
x⁺: streamwise coordinate in viscous wall units
y: wall-normal coordinate (m)
y⁺: wall-normal coordinate in viscous wall units
z: spanwise coordinate (m)
z⁺: spanwise coordinate in viscous wall units
γ: half-spot spreading angle (deg.)
Γ: circulation (⁠ $m^{2} / s$ ⁠)
$Γ'$: perturbation circulation (⁠ $m^{2} / s$ ⁠)
$δ_{0}^{*}$: boundary layer displacement thickness at the jet coordinate (mm)
$δ_{ω z}$: shear layer vorticity thickness (mm)
$ϵ^{+}$: nondimensionalized turbulence-kinetic-energy dissipation rate
ζ: coordinate that starts at the jet pipe outlet and increases toward its inlet (m)
$η^{+}$: Kolmogorov length scale in viscous wall units
θ_s: boundary layer momentum thickness at the cavity upstream edge (mm)
λ_z: streak spanwise spacing (m)
ν: kinematic viscosity (⁠ $m^{2} / s$ ⁠)
ρ: density (⁠ ${kg / m}^{3}$ ⁠)
τ: nondimensionalized time, $τ = t U_{0} / δ_{0}^{*}$
τ_w: wall shear stress
$ω'$: perturbation vorticity

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2021

Author(s)

Interaction of a turbulent spot with a two-dimensional cavity

I. INTRODUCTION

A. Physics of turbulent spots in boundary layers

B. Physics of the shear layer over a two-dimensional rectangular cavity

II. EXPERIMENTAL SETUP AND DATA PROCESSING

III. COMPUTATIONAL MODELS

A. Modeling in the Eulerian reference frame

1. Computational domain and boundary conditions

2. Base flow and turbulent spot initialization

3. Spatial grid

4. Solution method

B. Inviscid modeling in the Lagrangian reference frame

IV. FREESTREAM OPERATING CONDITIONS, BOUNDARY LAYER DEVELOPMENT, AND CAVITY FLOW DEVELOPMENT WITHOUT A TURBULENT SPOT

V. RESULTS AND DISCUSSION

A. Development of the turbulent spot upstream of the cavity

B. Development of the turbulent spot over and downstream of the cavity

1. Development of the leading edge region of the spot downstream of the cavity

2. Development of spot turbulence over and downstream of the cavity

3. Lagrangian modeling of the K–H roller distortion by the disturbance of a turbulent spot

4. Interaction of the turbulent spot with the cavity shear layer along the lateral boundaries of the spot

C. The wake region of the turbulent spot in the presence of feedback-driven oscillations

VI. CONCLUSIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

NOMENCLATURE

References

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Interaction of a turbulent spot with a two-dimensional cavity Open Access

I. INTRODUCTION

A. Physics of turbulent spots in boundary layers

B. Physics of the shear layer over a two-dimensional rectangular cavity

II. EXPERIMENTAL SETUP AND DATA PROCESSING

III. COMPUTATIONAL MODELS

A. Modeling in the Eulerian reference frame

1. Computational domain and boundary conditions

2. Base flow and turbulent spot initialization

3. Spatial grid

4. Solution method

B. Inviscid modeling in the Lagrangian reference frame

IV. FREESTREAM OPERATING CONDITIONS, BOUNDARY LAYER DEVELOPMENT, AND CAVITY FLOW DEVELOPMENT WITHOUT A TURBULENT SPOT

V. RESULTS AND DISCUSSION

A. Development of the turbulent spot upstream of the cavity

B. Development of the turbulent spot over and downstream of the cavity

1. Development of the leading edge region of the spot downstream of the cavity

2. Development of spot turbulence over and downstream of the cavity

3. Lagrangian modeling of the K–H roller distortion by the disturbance of a turbulent spot

4. Interaction of the turbulent spot with the cavity shear layer along the lateral boundaries of the spot

C. The wake region of the turbulent spot in the presence of feedback-driven oscillations

VI. CONCLUSIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

NOMENCLATURE

References

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Interaction of a turbulent spot with a two-dimensional cavity