Scaling and similarity laws in three-dimensional wall jets

Film cooling is a well-established technology to manage the temperature of solid components exposed to very hot streams. Film cooling is indispensable in modern gas turbine engines as it allows the operation of high-pressure turbine blades at gas temperatures far exceeding the melting temperature of the metal. Despite its numerous applications, few theoretical studies exist of the flow field of a film cooling device from a purely fluid mechanic point of view. The closest canonical flow to the flow produced by a film cooling device is a wall jet (Fig. 1). A wall jet is a flow produced by a slot or an orifice placed near a solid wall and ejecting fluid to flow near the wall in the direction parallel to the wall or by a jet impinging orthogonally on a wall. Wall jets and film cooling flows display features common to both jets and boundary layers.

The velocity profiles of wall jets are usually parameterized with respect to the maximum velocity $U m$ and its distance from the wall $y m$⁠, see Fig. 2(a). Especially in experimental studies, it is easier to identify the distance from the wall where the velocity is $U m / 2$⁠. This distance is customarily denoted as $y m / 2$ and is commonly used as a length scale for the study of wall jets. In wall-wakes and wall jets produced by film cooling devices, no local maximum of velocity can be identified. The velocity defect $Δ U$ with respect to the free-stream velocity $U e$ is also monotonic. In this case, it is convenient to base the definition of the wall jet length scale on the distance $δ 99$ where the mean velocity profile attains 99% of its free stream value $U e$⁠. In this case, the velocity defect $Δ U m$ evaluated at $y / δ 99 = 1 / 2$ is a suitable scale for the velocity profile. Diagrams showing the relation between the velocity profile and the quantities $Δ U m$ and $δ 99$ are shown in Figs. 2(b) and 2(c). For some film cooling jets, such as those produced by cylindrical films, the jet is lifted off the wall and a maximum velocity defect $Δ U m$ can be identified, see Figs. 2(d) and 2(e).

Early interest in film cooling and wall jets arose because of cooling needs in rocket engine combustion chambers.⁴ The first set of published wall jet measurements was performed by Förthmann,⁵ who referred to it as a “partially open jet.” Because of its implications on correlations for friction and heat transfer in cooling flows, the scaling of velocity and Reynolds stress profiles in wall jets is a matter of considerable interest and is the subject of a considerable body of the literature dating back to the years before World War II.

Over time, contrasting results have been proposed by a host of researchers. The current understanding is that the inner (near the wall) and outer (away from the wall) regions of a wall jet follow different scaling laws, but no broad consensus exists on the scaling of the Reynolds stress profiles or the terms in their budgets.

The following review summarizes some important results with a view to identify aspects of previous theoretical and experimental studies that can be used with advantage in a derivation of scaling laws for wall jets from prime principles.

Bradshaw and Gee¹⁵ studied wall jets within the context of boundary layer control and took hot wire measurements of jets flowing along flat and curved walls. They were the first to point out a major flaw in Glauert's solution, namely, the non-vanishing shear stress at the location of maximum velocity. They consequently took a stern view on the possibility of formulating numerical predictions for wall jets.

Kacker and Whitelaw^16,17 produced a detailed survey of wall jets and wall wakes based on hot-wire anemometry and published data on mean velocity profiles, turbulence intensities, Reynolds shear stress, spectra, correlation coefficients, and even selected terms in the turbulent kinetic energy (TKE) budgets, but did not attempt any form of normalization and did not discuss self-similarity properties exhibited by their results.

Significant contributions to the understanding of the evolution of wall jets at moderate distances from their injection slot were made by Sturgess.³ Sturgess studied the wall jet produced by cooling slots in combustion chambers with a view to correlating effectiveness¹⁸ data. Sturgess started from the observations that (i) most experimental studies are conducted on geometries not closely related to those found in actual combustors and (ii) data from different campaigns cannot be collapsed using commonly applied correlations. Sturgess attributed discrepancies to the existence of a potential core in the wall jet, which effectively offsets the start of the self-similar region from the geometric point of injection and produced a model to predict the potential core length. Sturgess^19–21 later revisited the issue of turbulence generation in cooling slots and correlated the performance of the slot with the extent to which turbulence and flow non-uniformity contribute to mixing in the early history of the jet itself. Turbulence and flow non-uniformity, in turn, was determined by the conditions in the plenum downstream of the metering holes.

Hammond²² produced a unified velocity profile for the inner and outer region of the wall jet in stagnant surroundings by merging Spalding's universal velocity profile²³ and Coles'²⁴ law of the wake. Hammond also derived a series of ancillary relations illustrating power laws between the main parameters of the flow, e.g., $U m$⁠, $y m$⁠, the downstream distance from the slot, and the local friction velocity. Hammond showed that the velocity profiles for a wall jet in inner coordinates do not comply to the same universal logarithmic law of the wall as equilibrium boundary layers.

Launder and Rodi²⁵ conducted a thorough review of the literature on measurements and modeling of wall jets up to 1983. Launder and Rodi summarized the state of understanding of turbulent wall jets up to that year by postulating scaling of the velocity profile with $U m$ and linear growth of $y m / 2$ with distance from the source of the jet.

Eriksson et al.²⁸ reported detailed measurements of mean and turbulent flow quantities and scaling parameters and reported confirmation of the mutual scaling of the maximum velocity and the jet layer thickness, with an exponent close to −1: $log ( U m ) ≈ − 1.083 − 0.573 log ( y 1 / 2 )$⁠. They also found a linear growth rate $y 1 / 2 / b = 0.0782 ( x / b ) + 0.332$⁠. Eriksson et al. also reported normal and shear Reynolds stresses normalized against the local maximum velocity in the wall jet, showing that full similarity is not attained for the Reynolds stresses in external coordinates.

George et al.²⁹ proposed a set of scaling laws based on the friction velocity $u *$ for the wall layer and the maximum velocity $U m$ for the outer layer based on the results by Eriksson et al. as well as other sources. They showed that the inner and outer layers are not as weakly coupled as initially postulated by Glauert⁷ and on the contrary a residual dependence on the length scale ratio $u * y 1 / 2 / ν$⁠. One of the interesting results obtained by George et al. is that the wall-normal Reynolds stress and the Reynolds shear stress scale with different quantities, namely, $U m$ and $u *$⁠. They also found that the maximum velocity must scale with an exponent no larger than $− 1 / 2$ with respect to downstream distance.

A recent contribution to the literature on wall jets was made by Gupta et al.,³¹ who derived a universal velocity profile for the buffer layer between the wall and the maximum velocity and showed that the correct coordinate for this part of the flow is rescaled by $R e τ$⁠.

Studies on three-dimensional jets started appearing in the literature later than studies on two-dimensional jets. Sforza and Herbst³² studied three-dimensional wall jets from rectangular orifices. They identified the effect of the jet potential core on the decay rate of the jet and showed the existence of two separate ranges for the spread of a three-dimensional wall jet and determined power laws describing the lateral and vertical growth of the jet, as well as the decay of the maximum velocity. The findings of Sforza and Herbst were reproduced by Kirkpatrick and Kenyon³³ a few years later but in the context of wall jets for air conditioning. Newman et al.³⁴ studied the three-dimensional wall jet issuing from a round hole. They found that the jet has a considerable lateral spread: The jet spreads laterally about seven times faster than it spreads in the direction normal to the wall. The mean velocity components attain self-preserving profiles, but the turbulence intensities do not, even if normalized by a local velocity scale. Swamy and Bandyopadhyay³⁵ conducted similar investigations. Swamy and Bandyopadhyay were able to identify not only the power-law spread regime and the radial-like spread regime but also the potential core of their jets. They found that the span-wise and wall-normal spread of the jet requires two separate virtual origins. They found no collapse of the turbulent quantities. A recent campaign of measurements in a three-dimensional wall jet was conducted by Namgyal and Hall,³⁶ who reported detailed measurements of mean velocity and six Reynolds stresses. Namgyal and Hall also reported estimates of individual TKE production terms and of the overall arrangement of the flow.

In their 1983 review, Launder and Rodi²⁵ examined the physical mechanism for the fast lateral spread of three-dimensional wall jets and tentatively identified streamline turning and Reynolds stress anisotropy as possible reasons. In a later contribution, Craft and Launder³⁷ examined the question of the lateral spread more in detail by attempting the computation of self-similar two-dimensional solutions based on Reynolds-stress transport models. They concluded that the axial vorticity and the lateral spread of the jet are driven by Reynolds stress anisotropy, very much like the secondary flow in a square duct. Craft and Launder produced flow visualizations closely resembling those produced by Sturgess¹⁹ in his study of turbulence at the exit of combustor cooling slots.

It is clear from this brief review that whereas some consensus exists on the scaling of mean velocity profiles in two- and three-dimensional wall jets, no such agreement exists on the scaling of turbulent quantities with local parameters and with the downstream distance. This paper derives such novel scaling laws from first-principles, invoking the conservation of mass, momentum, and energy fluxes for the outer part of the wall jet, neglecting the loss of momentum to the wall as a first approximation. The overall conservation equations are stated in Sec. II A. A small perturbation theory valid at large distances from the orifice is developed in Sec. II B, and estimates of growth rate, entrainment, and transversal velocity scales are derived in Sec. II C. The theory is corroborated by large eddy simulation (LES) data in Sec. III.

Consider the flow generated by a wall jet, such as a film cooling device. The wall jet is idealized as a rectangular region as shown in Fig. 3 where the fluid emitted by the film (the coolant) is partially mixed with the hot stream. The jet is surrounded by unperturbed coolant. The flow is developing along a wall and is subject to no imposed pressure gradients. The flow takes place predominantly in the streamwise direction in the hot stream. In the wall jet, the flow has components orthogonal to the free stream and these are assumed to be comparable in magnitude to the turbulent intensity.

The theory developed in Sec. II is now tested on a set of numerical results. The results represent a family of well-document film-cooling experiments. The experiments have been described in detail by Wittig et al.³⁸ The experiments represent single film cooling holes of cylindrical and fan-shaped cross section with an axis forming an angle of 30° to the direction of the wall. The ratio of hole length to diameter is $L / D = 6$⁠. The holes discharge into the main channel where a fully developed turbulent flow of hot gas is established. The Mach number in the main channel is $M e = 0.6$⁠, and the boundary layer thickness upstream of the injection location is $δ / D = 0.5$⁠. The experiments are conducted with blowing ratios $M = 0.5$⁠, $M = 1.0$⁠, and $M = 1.5$⁠. Further details about the experimental setup are reported in Thole et al.² and Gritsch et al.³⁹ 

The computations are based on a density-based code. The code is formally third-order accurate in time and space and adopts an implicit LES approach. The space discretization is cell-centered and uses MUSCL variable extrapolation to achieve spatial accuracy. Inviscid numerical fluxes are evaluated using Roe's Riemann solver. Inflow boundary conditions are imposed using a procedure developed by the authors⁴⁰ and reproducing realistic estimates of the two-point correlation tensor of the incoming turbulent flow. Full details of the numerical procedure, together with validation data based on the set of experiments by Wittig et al.,³⁸ are found in Hao and di Mare.⁴¹ For the purpose of the present calculations, a computational domain comprising the main channel, the hole, and its plenum is formed and discretized with multiblock grids. The grids are created starting with the plenum and the main channel. Stubs of the cooling film are added to the two domains, and the two stubs are joined at the common surface. The block layouts for the stubs are selected from templates according to the geometry of the film and its insertion angle. All grids are initially created using very coarse resolution. The grids used for the computations are obtained from these initial grids by successive rounds of refinement and smoothing with attraction toward the walls to guarantee sufficient near-wall resolution. The complete procedure is described by Hao et al.,⁴² Hao,⁴³ and Hao et al.⁴⁴ The domain extends 10D upstream of the injection location and 30D downstream. At the start of each computation, the LES simulation is run for around five flow-through times to eliminate the initial transient effects. Following that, statistics are collected for at least ten flow-through times for each case. The Courant number is 0.8.

The downstream evolution of the velocity defect $Δ U$ for the two types of films is shown in Fig. 4. The exponent $α$ is determined via least squares fit for the far-field of the two types of jet, and it is found to be $α = − 5 / 6$ for the fan-shaped film and $α = − 13 / 15$ for the cylindrical film. The values obtained from the analysis of the velocity defect data are, then, used to obtain the exponent of the power law for the jet layer thickness $b / D$ using Eq. (60). The power laws, thus, obtained are compared with the variation of $b / D$ obtained directly from the computations. In both cases, the agreement between the observed power law and the predicted power law based on the velocity defect data is excellent.

The mean flow profiles are shown for fan-shape and cylindrical films in Fig. 5. $η$ is the normalized distance from the wall. The profiles are extracted at distances between 5 and 20 orifice diameters downstream of the injection point and scaled according to the proposed scaling laws. The colored bands around the lines indicate the scatter of data taken on the jet centerline in the interval of $5 < ξ < 20$⁠. Narrow areas indicated the collapse of the data onto a single master curve, which represents a universal self-similar profile. The mean velocity defect profiles for the fan-shaped film show uniformly good collapse of all data between $x / D = 5$ and $x / D = 20$⁠, regardless of the blowing ratio. This finding is consistent with historical evidence on the collapse of wall jet and wall wake velocity profiles both in two and three dimensions. Similarly, good collapse can be obtained for the cylindrical film data, with the exception that it is now not possible to overlay data from different blowing ratios on a single curve. The data for wall-normal velocity in fan-shaped films corroborate the assumption in Eq. (28) that the normal velocity is small and, therefore, its contribution to the kinetic energy of the jet is small. It is also clear that the wall-normal velocity in the wall jet formed by a fan-shaped film is negative: This is a mere consequence of continuity and of the relatively high lateral spreading rate of the jet. In essence, the jet moves fluid from its upper side toward the wall and outwards toward its left and right edges. The collapse of wall-normal velocity data for the fan-shape film in Fig. 5 also shows that the exponent $2 − 3 α / 2$ for the scaling of the vertical velocity is correct. The kinematics of the cylindrical jet is qualitatively different. Here, the fluid is drawn upwards, away from the wall, whereas the footprint of the jet on the wall grows narrower as the jet evolves downstream.

In Fig. 5, data from the experiment reported in Thole et al.² are also analyzed based on the proposed scaling law and plotted for both the fan-shaped film and the cylindrical film. The experiment data are shown to collapse well with the simulated data. This additional data source can justify the proposed scaling law.

The Reynolds stress profiles are shown for fan-shape and cylindrical films in Fig. 6. The data in Fig. 6 need to be seen in the context of data for Reynolds stress in wall jets. The evidence so far on wall jets shows that the Reynolds stress data do not result in self-similar profiles when normalized with $U m 2$⁠. No systematic study exists of the self-similar properties of Reynolds stress profiles in film-cooling wall jets. Equation (56) shows that the Reynolds stress field in the wall jet of a film cooling device does not scale with the velocity defect squared. More precisely, Eq. (56) shows that when the Reynolds stresses or turbulence intensities are normalized by the velocity defect squared, the quantity obtained scales like $ξ − α$⁠. Some authors in the past have circumvented this difficulty by reporting Reynolds stresses normalized by the maximum value of the turbulent kinetic energy. The data in Fig. 6 show that by introducing the additional dependency on $ξ$ predicted by the proposed theory, the collapse of Reynolds stress data can be still obtained for fan-shaped films. The exponent $β = α$ is shown to be the correct exponent to rescale turbulence intensity and Reynolds stress. The data reported in Fig. 6 also show that for fan-shaped films the magnitude of the Reynolds stress field can be scaled with the blowing ratio.

The situation for cylindrical films, shown in Fig. 6, is slightly more complicated. In this case, no simple power law with respect to the blowing ratio M is suitable to rescale the Reynolds stress profiles. This happens because the different flow regimes—attached or lifted jets—represented in the current data impart different shapes to the Reynolds stress profile. Once the impossibility of collapsing curves from different blowing ratios is acknowledged, it is still possible to collapse data for each value of M separately. This is possible because the proposed theory can be applied to each flow condition separately. It is remarkable that, whereas different flow regimes result in different shapes of the Reynolds stress profiles, even for cylindrical films the algebraic relations binding the exponents $α$ and $β$ still hold and the values of this exponent are shown to be fairly insensitive to the blowing ratio.

Historical data on the wall jet show a clear consensus on the universality of mean velocity profiles and the downstream variation of the scaling parameters $U m$ and $y m / 2$⁠. No such consensus exists on the scaling of the turbulent quantities. The authors derive a set of scaling laws for the mean and turbulent flow quantities from prime principles. The first step in the derivation is to idealize the area occupied by the jet as a rectangular area. Mass, momentum, and energy fluxes are written in terms of uniform free-stream quantities and mean quantities within the jet. A set of relations is obtained describing the streamwise variation of streamwise velocity defect, density perturbation, span-wise and normal velocity, and turbulence intensity. It is shown that not all these quantities decay with the laws of the same power, but all can be expressed in terms of the exponent $α$ describing the decay of the velocity defect evaluated at a suitably chosen point along the jet layer thickness. One of the consequences of the scaling laws identified is that the correct scale for the turbulence intensity cannot be the velocity deficit squared. Furthermore, the shear stress at the wall is shown to be at most of the order $− 2 α$⁠. It is also shown that the large difference between span-wise and wall-normal growth rates observed in wall jets is a consequence of the conservation of momentum and energy, and of the initial hypothesis that the wall-normal velocities must be small. Numerical data indicate that the proposed theory is an accurate representation of the jet produced by a fan-shaped cooling film. It is also shown that for this type of device the wall-normal velocity profiles, velocity defects, and Reynolds stress profiles have shapes essentially independent of the blowing ratio M. A power law can also be identified that makes Reynolds stress data at various values of M collapse onto each other.

The jet produced by a cylindrical hole has a different flow arrangement. Nonetheless, power laws for the downstream development of velocity defect, transversal velocity scale, jet layer thickness, and turbulence intensity scale are derived in an identical fashion. The power laws obtained are shown to apply equally well to cylindrical films, albeit with modified exponents for the stream-wise decay of the velocity defect and growth of the jet layer thickness. It is seen that the velocity defect, wall-normal velocity, and Reynolds stress profiles have shapes sensitive to the blowing ratio M. As a consequence, it is not possible to identify a power law with respect to M that makes Reynolds stress profiles at various values of M collapse.

Rolls–Royce plc is gratefully acknowledged for supporting this work and for granting permission for its publication. The research is under Grant No. DFR03150. The authors sincerely thank Romero Eduardo, Frederic Goenaga, Cristian Orozco Pineiro, and Haidong Li from Rolls–Royce plc for helpful discussions. The authors also acknowledge the use of the University of Oxford Advanced Research Computing367 (ARC), the facility in carrying out this work. http://dx.doi.org/10.5281/zenodo.22558. Part of this work was also performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (Capital Grant No. EP/T022159/1) and DiRAC funding from the Science and Technology Facilities Council (www.dirac.ac.uk).

The authors have no conflicts to disclose.

Muting Hao: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Luca di Mare: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

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