A model of a turbulent boundary layer with a nonzero pressure gradient

Abstract

According to a model of the turbulent boundary layer that we propose, in the absence of external turbulence the intermediate region between the viscous sublayer and the external flow consists of two sharply separated self-similar structures. The velocity distribution in these structures is described by two different scaling laws. The mean velocity u in the region adjacent to the viscous sublayer is described by the previously obtained Reynolds-number-dependent scaling law φ = u/u_* = Aη^α, A = 1/ ln Re_Λ + 5/2, α = 3/2 In Re_Λ, η = u_*y/ν. (Here u_* is the dynamic or friction velocity, y is the distance from the wall, ν the kinematic viscosity of the fluid, and the Reynolds number Re_Λ is well defined by the data.) In the region adjacent to the external flow, the scaling law is different: φ = Bη^β. The power β for zero-pressure-gradient boundary layers was found by processing various experimental data and is close (with some scatter) to 0.2. We show here that for nonzero-pressure-gradient boundary layers, the power β is larger than 0.2 in the case of an adverse pressure gradient and less than 0.2 for a favorable pressure gradient. Similarity analysis suggests that both the coefficient B and the power β depend on Re_Λ and on a new dimensionless parameter P proportional to the pressure gradient. Recent experimental data of Perry, Marušić, and Jones were analyzed, and the results are in agreement with the model we propose.

Recent high-quality experiments by Perry, Marušić, and Jones (1–4) allowed us to clarify the model of the nonzero-pressure-gradient turbulent boundary layer. The model of the turbulent boundary layer at large Reynolds number proposed by Clauser (5) and Coles (6) is widely accepted and used. This model is based on the assumption that the transition from the wall region described by the Karman–Prandtl (7, 8) universal logarithmic law to the external flow is smooth. On the basis of our analysis of experimental data published over the last 30 years, we arrived at a different model (see refs. 9–11). According to our model, if the intensity of turbulence in the external flow is low, then the intermediate region between the viscous sublayer and the external flow consists of two self-similar structures separated by a sharp boundary. In particular, when the bilogarithmic coordinates lg φ, lg η are used, where φ = u/u_*, η = u_*y/ν, the mean velocity profile in the intermediate region has a characteristic form of a broken line (“chevron”) (Fig. 1). In part I of the intermediate region, the scaling law for the mean velocity distribution takes the form:

1

In part II, one finds a different scaling law:

2

The constants A, α, B, β can be determined with sufficient accuracy by processing the experimental data. According to our model, the expressions for A and α are identical to those in smooth pipes once the Reynolds number is defined correctly:

3

Here Re_Λ is an effective Reynolds number for a turbulent boundary layer, different from the usual Reynolds number Re_θ based on momentum thickness, which is arbitrarily although widely used in turbulent boundary layer studies. The test of the validity of our model is the closeness of two values of ln Re_Λ, ln Re₁ and ln Re₂, obtained by solving separately the two equations 3 with parameters A and α obtained from experimental data. Differences of less than 2–3% were obtained in all cases (see refs. 10 and 11 for previous data processing), and therefore we proposed to take ln Re_Λ as half the sum 1/2 (ln Re₁ + ln Re₂).

Figure 1.

Schematic representation of the mean velocity profile in developed the turbulent boundary layer in bilogarithmic coordinates lg (u/u*), lg (u*y/v).

In region II, the power β in the scaling law 2 for the zero-pressure-gradient boundary layer was found to be close to 0.2 (with some scatter). In cases of nonzero-pressure-gradient boundary layers, the values of β were found to be significantly different from 0.2. In the present Note, we perform a similarity analysis for the nonzero-pressure-gradient turbulent boundary layer. We find that both the coefficient B and the power β depend on Re_Λ and on a new similarity parameter P = ν∂_xp/ρu_*³. We compare the results of this analysis with experimental data by Marušić and Perry (1), Marušić (2), Jones, Marušić, and Perry (3), and Jones (4), and come to instructive conclusions.

The Model and the Similarity Analysis

According to our model, the turbulent boundary layer at large Reynolds numbers consists of two separate layers, I and II. The structure of the vorticity fields in the two layers is different, although both are self-similar. In layer I, the vortical structure is the one common to all developed wall-bounded shear flows, and the mean flow velocity is described by relations 1 and 3. In these relations, Re_Λ = UΛ/ν, where Λ is a characteristic length (12) close to 1.6 of the height of layer I.

The influence of viscosity is transmitted to the main body of the flow via streaks separating from the viscous sublayer.‖ The remaining part of the intermediate region of the boundary layer is occupied by layer II, where the relation 2 holds. It is well known [see in particular instructive photographs in Van Dyke (13)] that the upper boundary of the boundary layer is covered with statistical regularity by large-scale “humps,” and that the upper layer is influenced by the external flow via the form drag of these humps as well as by the shear stress. We have shown in earlier work that the mean velocity profile is affected by the intermittency of the turbulence, and as the humps affect intermittency, it is natural to see two different scaling regions. On the basis of these considerations, we have to determine a set of parameters that determine the coefficient B and the power β in 2. One of these parameters must be the effective Reynolds number Re_Λ, which determines the flow structure in layer I and is affected in turn by the viscous sublayer and by layer II. The following parameters should also influence the flow in the upper layer: the pressure gradient ∂_xp (x is the longitudinal coordinate reckoned along the plate; its origin is immaterial), the dynamic (friction) velocity u_*, and the fluid's kinematic viscosity ν and density ρ. The dimensions of governing parameters are as follows:

4

The first three have independent dimensions, so that only one dimensionless governing parameter can be formed:

5

Thus the parameters B and β should depend on the two parameters Re_Λ and P:

6

Comparison with Experimental Data

The data for nonzero-pressure-gradient flows are substantially less numerous than data for zero-pressure-gradient flows and do not yet allow us to construct surfaces B(Re_Λ, P), β(Re_Λ, P). However, the high-quality data obtained by Marušić and Perry (ref. 1, recently brought to completion via the Internet) and Jones, Marušić, and Perry (ref. 2, also completed on the Internet) allowed us to come to some instructive conclusions. The experiments of Marušić and Perry (1) were performed for two external flow velocities U: 10 m/s and 30 m/s. The experiments of Jones, Marušić, and Perry (3) were performed for three external flow velocities U: 5 m/s, 7.5 m/s, and 10 m/s. The results of the processing of the experimental data are presented in Table 1. Here x and Re_θ are given by the authors of the experiments, and Δ = 2| ln Re₁ − ln Re₂|/(ln Re₁ + ln Re₂).

Table 1.

Parameters of the scaling laws

	x, m	Reθ	α	A	β	B	ln Re1	ln Re2	ln ReΛ	Δ%
Data
Marušić	U = 10 m/s
	1.20	2,206	0.143	8.53	0.203	6.18	10.44	10.51	10.48	0.7
	1.80	3,153	0.150	8.30	0.227	5.45	10.05	10.03	10.04	0.2
	2.24	4,155	0.156	8.15	0.269	4.34	9.79	9.88	9.84	0.9
	2.64	5,395	0.171	7.54	0.345	2.87	8.73	8.77	8.75	0.5
	2.88	6,358	0.167	7.63	0.408	2.00	8.89	8.98	8.93	1.1
	3.08	7,257	0.169	7.57	0.450	1.64	8.78	8.88	8.83	1.2
	U = 30 m/s
	1.20	6,430	0.140	8.45	0.190	6.08	10.30	10.72	10.51	3.9
	1.80	8,588	0.145	8.41	0.207	5.63	10.24	10.32	10.28	0.8
	1.24	10,997	0.145	8.44	0.247	4.31	10.29	10.32	10.31	0.4
	2.64	14,208	0.147	8.39	0.306	2.91	10.20	10.20	10.20	0.1
	2.88	16,584	0.148	8.38	0.346	2.23	10.19	10.17	10.18	0.2
	3.08	19,133	0.145	8.45	0.388	1.71	10.31	10.35	10.33	0.4
Jones	U = 10 m/s
	0.18	855	0.144	8.39	0.20	6.36	10.21	10.45	10.33	2.4
	0.40	1,122	0.144	8.37	0.176	7.11	10.17	10.40	10.29	2.2
	0.60	1,314	0.146	8.28	0.168	7.41	10.01	10.25	10.13	2.4
	0.80	1,466	0.148	8.19	0.166	7.47	9.86	10.11	9.98	2.5
	1.00	1,616	0.144	8.38	0.160	7.68	10.19	10.44	10.31	2.5
	1.20	1,745	0.145	8.35	0.156	7.84	10.13	10.38	10.25	2.4
	1.40	1,888	0.142	8.44	0.153	7.99	10.29	10.55	10.42	2.5
	1.60	2,039	0.142	8.45	0.150	8.10	10.28	10.53	10.41	2.4
	1.80	2,150	0.143	8.41	0.148	8.18	10.23	10.50	10.36	2.6
	2.00	2,299	0.141	8.49	0.144	8.35	10.37	10.62	10.50	2.4
	2.20	2,411	0.144	8.37	—	—	10.17	10.43	10.30	2.5
	2.40	2,489	0.139	8.57	—	—	10.52	10.78	10.65	2.4
	2.60	2,574	0.145	8.32	—	—	10.08	10.36	10.22	2.7
	2.80	2,683	0.142	8.47	—	—	10.34	10.60	10.47	2.5
	2.92	2,728	0.145	8.31	—	—	10.06	10.33	10.19	2.7
	3.04	2,819	0.149	8.15	—	—	9.79	10.06	9.92	2.8
	3.16	2,832	0.147	8.24	—	—	9.94	10.20	10.07	2.6
	3.28	2,946	0.149	8.14	—	—	9.77	10.05	9.91	2.8
	3.40	2,987	0.142	8.46	—	—	10.32	10.60	10.46	2.7
	3.48	3,026	0.145	8.33	—	—	10.11	10.38	10.24	2.7
	3.54	3,032	0.146	8.29	—	—	10.03	10.30	10.16	2.7
	3.58	3,100	0.146	8.27	—	—	9.99	10.28	10.13	2.9
	3.62	3,029	0.147	8.20	—	—	9.88	10.20	10.04	3.2

For our subsequent analysis, we will use the series corresponding to U = 30 m/s of ref. 2 and U = 10 m/s of ref. 4 for the following reasons: in spite of a considerable variation in the usual parameter Re_θ, the effective Reynolds number Re_Λ obtained by the procedure we introduced remains nearly constant and close, for U = 30 m/s (2) to a constant ln Re_Λ = 10.3, and for U = 10 m/s (4) to a constant ln Re_Λ = 10.2. The mean velocity distribution in bilogarithmic coordinates for both series is presented in Fig. 2. Thus, we are able to obtain, with some approximation, cross sections of the surfaces B(Re_Λ, P), β(Re_Λ, P). The results corresponding to ln Re = 10.3 (adverse pressure gradient) are presented in Fig. 3 a and b; results corresponding to ln Re_Λ = 10.2 (favorable pressure gradient) are presented in Fig. 3 c and d. Note that for large values of the favorable pressure gradient, we were unable to reveal the second self-similar region. The situation is reminiscent of the disappearance of the second self-similar region in flows with an elevated level of free-stream turbulence. We found such a situation previously (10) when we processed the results of the remarkable experimental work of P. E. Hancock and P. Bradshaw (14).

Figure 2.

(A) The mean velocity profiles in bilogarithmic coordinates in the series of experiments of Marušić for U = 30 m/s; adverse pressure gradient. (1) Re_θ = 19,133, (2) Re_θ = 16,584, (3) Re_θ = 14,208, (4) Re_θ = 10,997, (5) Re_θ = 8,588, (6) Re_θ = 6,430. The “chevron” structure of the profiles is clearly seen, and regions I and II are clearly distinguishable. (B) The mean velocity profiles in bilogarithmic coordinates in the series of experiments of Jones for U = 10 m/s; favorable pressure gradient. (1) Re_θ = 855, (2) Re_θ = 1,122, (3) Re_θ = 1,314, (4) Re_θ = 1,616, (5) Re_θ = 2,728, (6) Re_θ = 3,032. The “chevron” structure of the profiles is clearly seen for curves 1–4, where β > α.

Figure 3.

(A) Cross section of the surface β(Re_Λ, P), for Re_Λ ≅ 10.3 (2). (B) Cross section of the surface B(Re_Λ, P), for Re_Λ ≅ 10.3 (2). (C) Cross section of the surface β(Re_Λ, P), for Re_Λ ≅ 10.2 (4). (D) Cross section of the surface B(Re_Λ, P), for Re_Λ ≅ 10.2 (4).

In the papers in refs. 1–4, the results concerning pressure were presented through a coefficient

where p_∞ is a constant reference pressure. Therefore, we calculated the parameter P by using the relation ∂_xp = 1/2ρU²∂_xC_p, where the density ρ canceled out; the values of all the other parameters are available in refs. 2 and 4. The values of the parameter P for U = 30 m/s (2) and for U = 10 m/s (4) are presented in Table 2.

Table 2.

Dimensionless parameters of the scaling law in region II

	Reθ	103 P	ln ReΛ
Data
Marušić	6,430	0	10.5
	8,588	1.75	10.3
	10,997	2.86	10.3
	14,208	4.2	10.2
	16,584	5.79	10.2
	19,133	7.04	10.3
Jones	855	−1.8	10.3
	1,122	−2.36	10.3
	1,314	−2.69	10.1
	1,466	−2.78	10.0
	1,616	−2.76	10.3
	1,745	−2.8	10.2

Eliminating the parameter P from relations 6, we obtain:

7

This relation is presented in Fig. 4 in the form of a dependence of B on 1/β. We see that this dependence is close to linear:

8

for the data by Marušić (2) (adverse pressure gradient) and

9

for the data by Jones (4) (favorable pressure gradient).

Figure 4.

(A) The dependence B(1/β) for Re_Λ ≅ 10.3; the straight line corresponds to 1.75/β − 2.8. (B) The dependence B(1/β) for Re_Λ ≅ 10.2; the straight line corresponds to 1/β + 1.43.

For layer I, there is also a linear relation between the coefficients A and 1/α, but contrary to B = B(1/β), this relation is universal. The coefficients in the relation B = B(1/β) should in principle depend on Re_Λ.

Conclusion

A new similarity parameter is obtained for the flow in the upper self-similar region of a developed nonzero-pressure-gradient turbulent boundary layer. Comparison with experimental data for nearly constant effective Reynolds numbers revealed simple (close to linear) Reynolds number-dependent relations between the parameters of the scaling law for the mean velocity distributions in the upper self-similar layer.

The investigation performed in the present Note and the papers in refs. 9–12 demonstrated that the Reynolds number-dependent scaling law for the velocity distribution across the shear flow obtained initially for flows in pipes is valid (with the same values of the constants) for the developed turbulent boundary layer flows. This allows us to expect that this scaling law reflects a universal property of all developed shear flows. The Reynolds number entering the law cannot be selected arbitrarily, for example, as Re_θ: it is uniquely determined by the flow itself. The simple procedure for the determination of the appropriate Reynolds number, which we proposed earlier, has been further validated in the present Note.

We expect that the same approach will work for more complicated flows: mixing layers, jets, and wall jets. However, the delicate task of investigating such flows requires high-quality experimental data, which are still lacking.

The concepts of incomplete similarity and vanishing viscosity asymptotics that we used for shear flows lead to plausible results for the local structure of developed turbulent flows. Here, however, high-quality experimental data are very rare, especially for the higher-order structure functions, where we have conjectured that divergences may occur.