Strouhal numbers associated with the drag crisis

Bartran, Dave

doi:10.1063/5.0250747

The supercritical Strouhal and critical Reynolds numbers associated with the drag crisis of circular cylinders in cross-flow are known to be dependent on surface roughness and turbulence levels. To better understand the role of free-stream turbulence, we consider the traditional model of oscillating lift forcing together with a randomly fluctuating disturbance associated with fluid turbulence. To the extent that these two components are additive, statistically independent, and Gaussian, we are able to derive a closed-form expression describing the relationship between the effective shedding rate and free-stream turbulence. A semi-empirical model of the critical Reynolds number accounting for fluid turbulence is also presented.

I. INTRODUCTION

Flow tests with subresonant cylinders, more commonly known as flexible cylinders with shedding rates well below resonance conditions,^1–5 have suggested that the reported supercritical Strouhal numbers associated with the drag crisis decrease in a near-linear fashion with increasing turbulence intensity, while another series of tests^6,7 have indicated a nonlinear increase.

In this paper, a simplified model of vortex shedding in the presence of turbulent velocity fluctuations is presented, and this is followed by an investigation of the integral scale length of the fluctuations to quantify its spectral content. Stochastic analysis is then applied to develop an expression for the effective shedding rate and its relationship with turbulence intensity and its spectral content.

In collecting the data associated with the drag crisis, we find that two categories of cylinders are involved. First, we have those with measurably smooth surfaces in pressurized wind tunnels and pipe flow tests,^1–5 and, second, those in atmospheric wind tunnel flow tests with less than ideal surfaces.^6,7 In the first case, with integral scale lengths of free-stream turbulence greater than the cylinder diameter, the supercritical Strouhal number is elevated over a significant range of Reynolds numbers at low turbulence, but decreases with increased turbulence intensity.^1–5 In the second case, flows with integral scales less than the cylinder diameter, the supercritical Strouhal number is increased in the presence of moderate turbulence level.^6,7 The critical Reynolds number data in both groups of tests are found to decrease with increasing turbulence intensity.

Occasionally, the supercritical shedding spectrum lacks an identifiable peak in reduced frequency coordinates.^4,5 When this occurs, we take the upper frequency half-power point of the oscillating lift spectrum as the effective Strouhal number⁵ that would be associated with resonance of more flexible or longer cylinders.

II. FLUID FORCING

Unlike earlier models involving vortex shedding in the presence of free-stream turbulence, such as that as in Ref. 8, we consider shedding rates well below the onset of flow-induced resonance. Avoidance of resonance is mandated in flow tests used to characterize Reynolds-number-dependent vortex shedding parameters^1–7 and the role played by free-stream turbulence.

The drag forces acting on a single cylinder in cross-flow with velocity fluctuations is given to first order by

F_{D} (t) = 0.5 ρ_{f} {[V + v^{'} (t)]}^{2} d C_{D} ≅ 0.5 ρ_{f} V^{2} [1 + \frac{2 v^{'} (t)}{V}] d C_{D} .

(1)

Extending this to the oscillating lift forcing due to vortex shedding and to that arising from the spatial–temporal distribution of turbulent eddies, we have the combined lift forcing

{\tilde{F}}_{L} (t) = \frac{F_{L} (t)}{0.5 ρ_{f} V^{2} d} ≅ c_{L} \sin [ω_{0} t + ϕ (t)] + c_{R} \frac{δ V (t)}{V} .

(2)

The Reynolds-number-dependent forcing parameters are (c_L, c_R), with low turbulence shedding rate ω₀, uniformly distributed phase disturbances ϕ(t), and zero mean velocity fluctuations v′(t). This model of vortex shedding with additive turbulent forcing is analogous to those suggested previously, although in spectral form.^9,10 The phase disturbances are considered to be independent of the velocity fluctuations. Were ϕ(t) deterministic, its rate of change would act to modulate the effective shedding rate directly.

The root mean square (RMS) of the fluctuations, normalized by the mean fluid velocity, defines the turbulence intensity as $TI \equiv \sqrt{⟨v^{'} {(t)}^{2}⟩} / V$ ⁠, while the spectral bandwidth of the fluctuations is inferred from reported integral scale lengths using autocorrelation methods.^5,9–11 The magnitude of the turbulence forcing parameter may be estimated by examining the changes in the oscillating lift parameter that result from increased turbulence intensity, as in Refs. 6 and 7. As most flow tests lack such data, the forcing parameter ratio c_R/c_L, is considered an adjustable parameter.

III. TURBULENCE SPECTRA

According to the Taylor hypothesis,^5,10 the integral scale length of free-stream turbulence is related to the mean fluid velocity and the integral time scale of its fluctuations

Λ ≅ V τ_{Λ} .

(3)

In Ref. 12, the following normalized turbulence spectra is suggested

h (ω) ≅ \frac{1}{1 + {(ω / ω_{BW})}^{2}},

(4)

but this has been found to lead to problems when one needs to calculate the variance of its rate of change, as in Sec. IV. As a consequence, we shall rely on a bandlimited white noise model of free-stream turbulence to highlight the importance of its spectral bandwidth in altering the shedding rates in both flow tests and simulations.

The autocorrelation of the turbulence spectra is given by its inverse Fourier cosine transform.¹¹ In the case of bandlimited white noise, we begin with the normalized power spectra g(ω), which is unity within the defined bandwidth 0 ≤ ω ≤ ω_BW and zero otherwise. The autocorrelation is now

R (τ) = \frac{2}{π} \int_{0}^{ω_{BW}} g (ω) \cos (ω τ) d ω = \frac{2 \sin (ω_{BW} τ)}{π τ} .

(5)

In cases where Eq. lacks a well defined zero-crossing, the exponential method^13,14 offers a lower bound estimate of the time scale. For the present, we shall integrate Eq. from the origin to the first zero-crossing^5,11 τ₀ = π/ω_BW. We now have an integral time scale of

τ_{Λ} ≅ R^{- 1} (0) \int_{0}^{τ_{0}} R (τ) d τ = \frac{1.859}{ω_{BW}} .

(6)

On combining Eqs. and , we find that the turbulence bandwidth is now ω_BW = 2πV/Λ. Normalizing the turbulence bandwidth by the low turbulence shedding rate ω₀ = 2πSt₀V/d, at the fluid velocities being considered, together with the low turbulence Strouhal number St₀, we have a bandwidth parameter defined as

α \equiv \frac{ω_{BW}}{ω_{0}} ≅ \frac{0.296}{S t_{0} Λ / d} .

(7)

This parameter proves useful in interpreting Strouhal number behavior in a given Reynolds number range. It also addresses the question raised by Zdravkovich¹⁵ regarding the lack of adequate spectral details of free-stream turbulence,⁶ where the Strouhal numbers are found to increase with increasing turbulence. Unlike models and tests involving flow-induced resonances,^8,16,17 Eq. (7) is defined in terms of the turbulence-free shedding rate of a subresonant cylinder to facilitate identification of potential resonance conditions that must be avoided at higher velocities.

In Sec. IV, we address the effective shedding rate in the presence of free-stream turbulence. This is accomplished by using random noise theory to quantify the zero-crossing rate of the shedding forces, and hence the effective Strouhal number. While the theory can be applied in other Reynolds regimes, the present focus is on Strouhal numbers associated with the drag crisis.

IV. THE ZERO-CROSSING PROBLEM

Traditional forcing models¹⁷ are used routinely for engineering risk assessments of intrusive pipe fittings. However, consideration of elevated, supercritical Strouhal numbers with isolated cylinders has remained controversial, in spite of documented failures, and subsequent flow tests involving the drag crisis.^5,18 To better understand the role of free-stream turbulence in such cases, consider the following.

When the leading term in Eq. (2) is characterized by stationary random fluctuations in magnitude with mean frequency and uniformly distributed phase disturbances {0, …, 2π}, it is formally described by Gaussian statistics.^19–21 Under steady flow conditions, the central limit theorem commonly applies, and oscillating lift forcing may be considered as Gaussian, if only as an approximation.

When the oscillating lift term is combined with independent normally distributed turbulent forcing, the effective shedding rate may be quantified by forming the joint probability of the zero-crossings of the combined forcing and its rate of change.¹⁹ We begin by defining the joint probability of the zero-crossing rate given by

Ψ (\tilde{α}, \tilde{β}) = \frac{1}{2 π σ_{\tilde{F}} σ_{\tilde{F^{'}}}} \exp [- \frac{1}{2} (\frac{{\tilde{α}}^{2}}{σ_{\tilde{F}}^{2}} + \frac{{\tilde{β}}^{2}}{σ_{\tilde{F^{'}}}^{2}})],

(8)

with level crossing conditions

{\tilde{α}, \tilde{β}}

and variances of the combined forcing

{σ_{\tilde{F}}^{2}, σ_{\tilde{F^{'}}}^{2}}

⁠.

First we develop the individual variance components {s²(t), n²(t)} and their respective rates of change {s′²(t), n′²(t)} for the bandlimited white noise model of free-stream turbulence. Thus,

\begin{gathered} ⟨s^{2} (t)⟩ = \frac{1}{2 π} \int_{0}^{2 π} c_{L}^{2} \sin^{2} [ω_{0} t + ϕ (t)] d ϕ = \frac{c_{L}^{2}}{2}, \\ ⟨{s^{'}}^{2} (t)⟩ = \frac{1}{2 π} \int_{0}^{2 π} {(ω_{0} c_{L})}^{2} \cos^{2} [ω_{0} t + ϕ (t)] d ϕ = \frac{{(ω_{0} c_{L})}^{2}}{2} \end{gathered}

(9)

and

\begin{gathered} ⟨n^{2} (t)⟩ = {(TI c_{R})}^{2} \int_{0}^{\infty} g (ω) d ω = {(TI c_{R})}^{2} ω_{BW}, \\ ⟨{n^{'}}^{2} (t)⟩ = {(TI c_{R})}^{2} \int_{0}^{\infty} ω^{2} g (ω) d ω = \frac{{(TI c_{R})}^{2} ω_{BW}^{3}}{3} . \end{gathered}

(10)

We now have the required variances of the combined forcing and its rate of change:

σ_{\tilde{F}}^{2} = \frac{c_{L}^{2}}{2} + {(TI c_{R})}^{2} ω_{BW}, σ_{{\tilde{F}}^{'}}^{2} = \frac{{(ω_{0} c_{L})}^{2}}{2} + \frac{{(TI c_{R})}^{2} ω_{BW}^{3}}{3},

(11)

Combining Eqs. and and integrating over all rates of change yields the zero-crossings of the oscillating lift forcing per unit time:

N_{0}^{'} = {\lim_{Δ \to \infty} \int_{- Δ}^{Δ} |\tilde{β}| Ψ (\tilde{α}, \tilde{β}) d \tilde{β}|}_{\tilde{α} = 0} = {\frac{1}{π} \frac{σ_{\tilde{F^{'}}}}{σ_{\tilde{F}}} \exp (- \frac{0.5 {\tilde{α}}^{2}}{σ_{\tilde{F}}^{2}})|}_{\tilde{α} = 0} .

(12)

Recognizing that we have two such zero-crossings per cycle, combining Eqs. and and using the Strouhal relationship allows us to write the turbulence-modified Strouhal number as

S t (TI) = S t_{0} {[\frac{1 + \frac{2}{3} {(TI c_{R} / c_{L})}^{2} α^{2} ω_{BW}}{1 + 2 {(TI c_{R} / c_{L})}^{2} ω_{BW}}]}^{1 / 2},

(13)

where St₀ is the turbulence-free Strouhal number at the Reynolds numbers associated with the drag crisis, α is the bandwidth parameter, and ω_BW is the turbulence bandwidth.

The partial derivative of Eq. (13), ∂St/∂TI, is proportional to (α² − 3), revealing a bifurcation or branching in the shedding rate dependence on TI. As a result, we have a decreasing St(TI) for α² < 3, and increasing St(TI) for α² > 3, with St(TI) independent of TI for α² = 3. When higher-order integrable spectra are considered, both the bandwidth parameter and the branching condition of Strouhal number behavior are altered with slightly improved data fits in Fig. 1.

It is readily shown that the steady-state energy transfer between the fluid forcing and the cylinder is proportional to the shedding rate. As a consequence, both experimentally and within the framework of the model presented here, we find that turbulence causes a decrease in the shedding rate for α² < 3 and an increase for α² > 3.

V. FLOW TEST DATA AND MODEL PREDICTIONS

Figure 1 compares the supercritical Strouhal numbers^1–7 with the results of Eq. (13) using the Strouhal number data associated with the drag crisis in each test together with the respective turbulence intensities and reported integral scale lengths. The data in Figs. 1–3 are summarized in Table I.

FIG. 1.

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Supercritical Strouhal number vs TI for two datasets. The dashed lines represent Eq. (13) for the parameters shown and the solid lines highlight the datasets involved.

FIG. 2.

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Turbulence bandwidth parameters vs TI using reported data to calculate the bandwidth parameter according to Eq. (7).

FIG. 3.

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Mean critical Reynolds numbers vs TI using drag force measurements.

TABLE I.

Collected data used in Figs. 1–3.

Reference	TI (%)	Λ/d	α	St(0)	St(TI)	⟨Rd_crit⟩
Bearman²	0.20	⋯	⋯	0.46	0.46	346 410
Schewe¹	0.40	⋯	⋯	0.48	0.48	324 037
Achenbach and Heinecke³	0.46	⋯	⋯	0.51	0.51	273 861
So and Savakar⁴	0.50	⋯	1.18	0.50	0.49	256 905
Kawamura et al.⁵	1	5.7	0.11	0.48	0.48	223 607
Kawamura et al.⁵	5	0.53	1.18	0.48	0.42	146 969
So and Savakar⁴	9.5	⋯	1.18	0.23	0.34	60 000
Kawamura et al.⁵	13	0.089	7.93	0.48	0.281	63 246
Blackburn and Melbourne⁷	18	0.53	2.43	0.23	0.23	⋯
Cheung and Melbourne⁶	0.4	0.521	2.47	0.23	0.23	229 169
Cheung and Melbourne⁶	1.6	0.521	2.47	0.23	0.25	141 421
Cheung and Melbourne⁶	4.4	0.521	2.47	0.23	0.30	109 545
Cheung and Melbourne⁶	6.8	0.521	2.47	0.23	0.32	69 282
Cheung and Melbourne⁶	9.1	0.521	2.47	0.23	0.34	77 460

Reference	TI (%)	Λ/d	α	St(0)	St(TI)	⟨Rd_crit⟩
Bearman²	0.20	⋯	⋯	0.46	0.46	346 410
Schewe¹	0.40	⋯	⋯	0.48	0.48	324 037
Achenbach and Heinecke³	0.46	⋯	⋯	0.51	0.51	273 861
So and Savakar⁴	0.50	⋯	1.18	0.50	0.49	256 905
Kawamura et al.⁵	1	5.7	0.11	0.48	0.48	223 607
Kawamura et al.⁵	5	0.53	1.18	0.48	0.42	146 969
So and Savakar⁴	9.5	⋯	1.18	0.23	0.34	60 000
Kawamura et al.⁵	13	0.089	7.93	0.48	0.281	63 246
Blackburn and Melbourne⁷	18	0.53	2.43	0.23	0.23	⋯
Cheung and Melbourne⁶	0.4	0.521	2.47	0.23	0.23	229 169
Cheung and Melbourne⁶	1.6	0.521	2.47	0.23	0.25	141 421
Cheung and Melbourne⁶	4.4	0.521	2.47	0.23	0.30	109 545
Cheung and Melbourne⁶	6.8	0.521	2.47	0.23	0.32	69 282
Cheung and Melbourne⁶	9.1	0.521	2.47	0.23	0.34	77 460

The dimensionless turbulence forcing parameter c_R/c_L for the two groups of flow tests is adjusted for nominal fits with the data as shown. The Strouhal number behavior for α²≅ 3, commonly associated with extreme surface roughness³ or with localized turbulence,²² is also seen as an asymptotic limit of the Strouhal number data at extreme turbulence,^5,7 as in cross-flow heat exchangers.

In the following, we consider three classes of turbulence intensity: low turbulence (TI < 1%), moderate turbulence (TI < 5%–8%), commonly present in straight pipe flow, and extreme turbulence (TI > 10%), found in highly disturbed flows adjacent to piping elbows, heat exchangers, turbines, and combustion processes. The role played by surface roughness will become more apparent when we examine the critical Reynolds number data.

Representative supercritical Strouhal numbers vs TI are shown in Fig. 1 for the two groups of flow tests and are clearly distinguished by their bandwidth parameters at low to moderate TI. At high levels of turbulence, the two datasets appear to merge, followed by decreasing Strouhal numbers with increased turbulence, regardless of the bandwidth parameter. This apparent merger marks the onset of turbulence-dominated forcing. For example, arbitrarily considering a doubling of the relative RMS forcing as marking the onset of turbulence-dominated forcing or $σ_{\tilde{F}} (TI) / σ_{\tilde{F}} (0) = 2$ ⁠, turbulence domination is expected to occur at TI ≅ 10% and TI ≅ 6%, respectively, for the two datasets in Fig. 1. As only subresonant cylinders are tested, the displacement is linearly related to the applied force, and so the forcing and displacement ratios are identical. The RMS error of the data fit for the two datasets is 3%.

The representative supercritical Strouhal numbers shown in Fig. 1 suggest the following. Narrowband turbulence, presently defined as α² < 3, tends to interfere with the shedding process, with a reduction in the shedding rate with increasing turbulence intensity.⁵ Broadband turbulence, however, with α² > 3, tends to energize the shedding process⁶ at low to moderate turbulence, with the shedding rate increasing with increasing turbulence intensity. At extreme levels of TI, the shedding rate in both cases decreases, approaching that of subcritical Reynolds numbers or highly roughened surfaces.

Figure 2, and also Table I, show the bandwidth parameters using the reported St₀ and longitudinal integral scales Λ/d at velocities well into the drag crisis.

At low to moderate turbulence levels, the datasets are separated by the Strouhal bifurcation, but for TI greater than 8%–10%, we find a combination of elevated bandwidth parameters with a turbulence-dominated reduction in Strouhal numbers^5,7 for both liquid and atmospheric wind tunnel tests.

Finally, the mean critical Reynolds numbers determined from reported drag force measurements, in the presence of free-stream turbulence, are shown in Fig. 3, with error bars marking the transition associated with the drag crisis.

Considering the critical Reynolds number as a threshold condition at zero turbulence, the turbulence-modified critical Reynolds number must satisfy

R d_{crit} (0) \geq R d_{crit} (TI) (1 + \frac{| v^{'} (t) |^{peak}}{V}) .

(14)

Since the velocity fluctuations are characterized by the variances of their power spectra, Eq. , we have

σ^{2} = T I^{2} \int_{0}^{ω_{BW}} g (ω) d ω = T I^{2} ω_{BW},

(15)

with velocity amplitudes proportional to the RMS,

TI \sqrt{ω_{BW}}

⁠. Rearranging Eq. and including an empirical factor A, we have

R d_{crit} (TI) ≅ \frac{R d_{crit} (0)}{1 + A TI \sqrt{ω_{BW}}} .

(16)

This expression is consistent with the finding²³ that turbulence with larger spectral bandwidth is more effective in reducing Rd_crit(TI).

The flow test data used to solve these equations in Fig. 3 are Rd_crit(TI) and TI. Taking two points in each dataset, one at low turbulence (TI < 1%) and the second at the onset of extreme turbulence (TI ≅ 8%–13%), and solving for Rd_crit(0) and A, we have the following trends, as shown in Fig. 3, with RMS fitting errors of 20%:

R d_{crit} (TI) ≅ \{\begin{cases} 372 000 / (1 + 0.376 TI %) & for α < 3, \\ 252 000 / (1 + 0.247 TI %) & for α > 3 \cdot \end{cases}

(17)

The reduction in critical Reynolds numbers for the broadband dataset in Fig. 3 is found to be the result of the actual surface conditions of the cylinders tested,⁷ as noted in subsequent surface roughness measurements.²⁴

VI. CONCLUSION

Application of Bendat’s analysis¹⁹ to vortex shedding was motivated by a study of thermowell failure in pipe flow,²⁵ where consideration of a range of Reynolds-number-dependent supercritical Strouhal numbers is required for proper risk assessment. While this is hardly a solution of the Navier–Stokes equations, it helps resolve various interpretations of the supercritical Strouhal numbers associated with wind-induced vortex shedding rates^6,7,24 of architectural and engineering structures, whether due to turbulence parameters, surface roughness, or combinations thereof.

The use of thermowell calculations based on traditional sinusoidal forcing models is still possible. First, the Reynolds numbers should be checked under maximum flow conditions. If the fitting is exposed to supercritical Reynolds numbers, then two calculations should be performed over the expected flow range, one with and one without elevated Reynolds-number-dependent Strouhal numbers, as in Refs. 17, 25, and 26. Should potential resonance conditions be identified, a reduction in the length of the intrusive fitting will often eliminate the risk of resonance. Such changes have little effect on measurement error or speed of response, as can be calculated analytically or by simulation.

Subject to additional flow tests and simulations that account for fluid turbulence, preliminary evaluations suggest that the proposed model with suitable Reynolds-number-dependent forcing parameters may be used in characterizing the Strouhal number for subcritical Reynolds numbers.²⁷ The theory may also provide additional insight into the application of straked thermowells,²⁸ which exhibit contrasting behavior to that observed in tests with roughened cylinders,^3,29 since weakened resonances are observed, with elevated Strouhal numbers marking the postcritical Reynolds transition in high-pressure low-turbulence gas flows.³⁰ The analysis presented here also explains the behavior of vortex flow meters³¹ and in so doing suggests that errors associated with free-stream turbulence may be corrected with suitable flow conditioners.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Dave Bartran: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that supports the findings of this study are available within the article.

NOMENCLATURE

c_L: mean RMS oscillating lift force parameter
c_R: mean RMS turbulence forcing parameter
d: cylinder diameter
$\tilde{F}, \tilde{F^{'}}$: dimensionless forcing and its rate of change
g(ω): single-sided bandlimited white noise spectra
h(ω): single-sided higher-order monotonic spectra
$N_{\tilde{α}}^{'}$: level-crossing rate (twice the effective frequency)
Rd: Reynolds number, Vd/ν
St: Strouhal number, (ω/2π) d/V
St₀: Strouhal number at minimal turbulence
TI: turbulence intensity, $\sqrt{⟨{v^{'}}^{2} (t)⟩} / ⟨V⟩$
V, v′(t): mean fluid velocity and its fluctuations in time

Greek

α ≡ ω_BW/ω₀: turbulence bandwidth parameter
$\tilde{α}, \tilde{β}$: level-crossing conditions for forcing and its rate of change
Λ: integral scale length of turbulent fluctuations
ρ_f: fluid density
$σ_{\tilde{F}}^{2}, σ_{\tilde{F^{'}}}^{2}$: variance of lift forcing and of its rate of change
τ_Λ: integral time scale of turbulent eddies
ϕ(t): uniformly distributed phase fluctuations of sinusoidal forcing
ω: frequency (rad/s)
ω₀: vortex shedding rate at minimal turbulence
ω_BW: spectral bandwidth of velocity fluctuations

NOMENCLATURE

c_L: mean RMS oscillating lift force parameter
c_R: mean RMS turbulence forcing parameter
d: cylinder diameter
$\tilde{F}, \tilde{F^{'}}$: dimensionless forcing and its rate of change
g(ω): single-sided bandlimited white noise spectra
h(ω): single-sided higher-order monotonic spectra
$N_{\tilde{α}}^{'}$: level-crossing rate (twice the effective frequency)
Rd: Reynolds number, Vd/ν
St: Strouhal number, (ω/2π) d/V
St₀: Strouhal number at minimal turbulence
TI: turbulence intensity, $\sqrt{⟨{v^{'}}^{2} (t)⟩} / ⟨V⟩$
V, v′(t): mean fluid velocity and its fluctuations in time

Greek

α ≡ ω_BW/ω₀: turbulence bandwidth parameter
$\tilde{α}, \tilde{β}$: level-crossing conditions for forcing and its rate of change
Λ: integral scale length of turbulent fluctuations
ρ_f: fluid density
$σ_{\tilde{F}}^{2}, σ_{\tilde{F^{'}}}^{2}$: variance of lift forcing and of its rate of change
τ_Λ: integral time scale of turbulent eddies
ϕ(t): uniformly distributed phase fluctuations of sinusoidal forcing
ω: frequency (rad/s)
ω₀: vortex shedding rate at minimal turbulence
ω_BW: spectral bandwidth of velocity fluctuations

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This work is licensed under a Creative Commons Attribution 4.0 International License .

Strouhal numbers associated with the drag crisis

I. INTRODUCTION

II. FLUID FORCING

III. TURBULENCE SPECTRA

IV. THE ZERO-CROSSING PROBLEM

V. FLOW TEST DATA AND MODEL PREDICTIONS

VI. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

NOMENCLATURE

Greek

NOMENCLATURE

Greek

REFERENCES

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Strouhal numbers associated with the drag crisis Open Access

I. INTRODUCTION

II. FLUID FORCING

III. TURBULENCE SPECTRA

IV. THE ZERO-CROSSING PROBLEM

V. FLOW TEST DATA AND MODEL PREDICTIONS

VI. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

NOMENCLATURE

Greek

NOMENCLATURE

Greek

REFERENCES

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Strouhal numbers associated with the drag crisis