1 Introduction

Shock wave-turbulent boundary layer interaction (SWTBLI) is one of the most ubiquitous yet perplexing phenomena occurring in a wide variety of internal and external aerodynamics applications. This phenomenon could lead to significantly negative effects on the performance of high-speed vehicles and turbomachines, e.g., increasing the surface drag and thermodynamical loading and inducing the flow separation. Simultaneously, the unsteady motion of both the shock system and the separation bubble accompanies the SWTBLI, which operates in an extremely low frequency in comparison with the coherent structure of the turbulence and hence causes an increment in the complexity^[1].

Through the experiments on a variety of models associated with the boundary layers with laminar, transitional, and turbulent separations, Chapman et al.^[2] proposed an early remarkable concept, i.e., free interaction. It was found that the pressure rise to separation is independent of the inducing separation mode for either laminar or turbulent separation in supersonic flows. The plateau pressure rise is also independent for laminar separation, but it significantly depends on the model geometry for turbulent separation. The first asymptotic description of a supersonic laminar boundary layer through a free interaction was provided independently by Stewartson and Williams^[3] and Neiland^[4]. According to their arguments, in the close neighborhood O(Re^3/8L) of the shock/boundary-layer interaction region, three layers, which are expressed in terms of the magnitude O(Re^3/8L) for the upper deck, the magnitude O(Re^1/2L) for the main deck, and the magnitude O(Re^5/8L) for the lower deck, emerge in the wall-normal direction when Re → ∞, where Re is the Reynolds number based on the distance from the local interaction location to the leading edge L. A series of subsequent work, as reviewed by Adamson and Messiter^[5], has depicted the whole scenario of the interactions of the shock waves with laminar boundary layers. It has been revealed that, under the laminar conditions, the heat transfer rates, as well as other dynamical values, even with the high temperature effects, can be predicted by the computational fluid dynamics (CFD) accurately with adequate meshes if the freestream conditions are properly characterized (see Ref. [1]).

However, the investigations on the interactions with the turbulent boundary layers are insufficient. This may be due to that the numerical predictions from the Reynolds averaging Navier-Stokes (RANS) equations with turbulence modeling are unsatisfactory and sometimes disastrous, especially on predicting the heat flux and surface drag^{[1, 5-6]}.

Extensive experimental and numerical investigations^[7-17] have been performed to provide comprehension on the relative physical features and serve as the databases for CFD validation.

One factor that affects the accuracy of the CFD prediction attributes to the uncertainty of the temperature boundary condition at the wall. As pointed out by Edwards^[18], in some cases, a pre-test ambient temperature is designated as the wall temperature of the CFD calculation, but the run time of the wind tunnel is long enough to render this assumption questionable. Also, the SWTBLIs often result in an order of magnitude enhancement on the heat transfer rates, which leads to locally high-temperature phenomena. Therefore, it is rational to take the real-gas effect and the possible chemical reaction into account. However, as illustrated by Knight et al.^[19], the applied chemical models would be crucial to the numerical results, which reinforces the uncertainty of the numerical prediction. Even when the comparisons are made between the RANS solutions and those from the direct numerical simulations (DNSs) or large eddy simulations (LESs), salient discrepancies are always found both qualitatively and quantitatively^[17].

The rational improvement on the accuracy of the turbulence models on predicting SWTBLIs requires further understanding on the features of the flow motion, especially the near-wall behaviors. Loginov et al.^[20] examined the development of the mean-velocity profiles of a supersonic flow past a compression ramp from the LES solution. A turbulent-laminar reverse transition or relaminarization in the reverse-flow region is observed. It is well-known that for a turbulent boundary layer with zero pressure gradient (ZPG), the mean-velocity profiles in the near-wall region satisfy the linear law in the viscous sublayer and the logarithmic law in the mesolayer. Such behaviors are found to be independent of the flow conditions such as the Mach number, the Reynolds number, and the temperature conditions at the wall (see Wilcox^[21] and Duan et al.^[22]). However, with the presence of a sufficiently strong adverse pressure gradient (APG), the traditional laws of the wall cease to be valid. As proposed by Skote and Henningson^[23] from their study on the near-wall behavior of the incompressible turbulent boundary layer with separation, the mean-velocity is described by a pair of new laws corresponding to two sublayers. The new relations require the balance among the convection, the shear stress (including both the viscous and Reynolds stresses), and the pressure gradient, and are proven to be adequate via the DNS data. The normalization adopted in the new relations are also able to reproduce the Reynolds stresses in the incompressible turbulent boundary layer with separation (see Gungor et al.^[24]).

The objective of the present paper is to shed some additional light on the near-wall behavior of the interactions in the SWTBLI. The laws of the wall as proposed by Skote and Henningson^[23] for the incompressible separation are followed, but the compressible effect is considered additionally due to the supersonic nature.

2 Numerical methods 2.1 Physical model

In order to demonstrate the physical model to be studied in this paper, we plot the contours of the instantaneous temperature on a spanwise slice (see Fig. 1). We focus on the interaction between an impinging oblique shock wave and a turbulent boundary layer on a flat plate. The Cartesian coordinates (x, y, z), which, respectively, denote the streamwise, wall-normal, and spanwise directions, are employed, in which the reference length δ_in is selected as the momentum thickness of the boundary layer at the inlet of the computational domain.

It is clear from Fig. 1 that, the flow remains laminar in the very early stage. Since the artificial perturbation with the form of blowing and suction at the wall is introduced near the inlet, the laminar-turbulent transition emerges at about x=220. The turbulence develops further downstream until it reaches the outlet of the computational domain. Moveover, an oblique shock wave is introduced from the inviscid region, and impinges on the turbulent boundary layer at about x=420, which generates even higher level fluctuations in the turbulent boundary layers. Simultaneously, a reflect shock wave is observed downstream of the interactive region. This interaction is often accompanied with flow separation and low-frequency unsteadiness, which challenges the accuracy of turbulence modeling prediction and consequently is of our interest.

2.2 Numerical methods

The governing equations are the three-dimensional Navier-Stokes equations for a perfect gas, which can be found in Qin and Dong^[25]. The DNS code is kindly provided by Prof. Xinliang LI of Institute of Mechanics, Chinese Academy of Science. The code has been extensively used for a variety of simulations involving turbulent boundary layers and shock waves^[26-28].

All the physical quantities are normalized by the freestream quantities, i.e.,

where ρ, u_i (i=1, 2, 3), T, p, and μ denote the density, the velocities, the temperature, the pressure, and the dynamic viscosity, respectively, the superscript * represents the dimensional quantities, and the subscript ∞ represents the dimensional quantities at the freestream. The Mach number is defined by Ma_∞ = u_∞/c_∞, and the Reynolds number is defined by Re = ρ_∞u_∞δ_in/μ_∞, where c_∞ is the sound speed.

The seventh-order weighted essentially nonoscillatory (WENO) scheme^[9] is used for the nonlinear convection terms, and the eighth-order central scheme is used for the viscous terms. The time integration is performed by use of an explicit forth-order Runge-Kutta scheme.

2.3 Computational setup

The computational parameters for the simulation of the SWTBLI conducted in this paper are listed in Table 1. A blowing and suction slot between x=20 and x=40 (see Fig. 1) is introduced as a perturbation generator, which subsequently leads to the emergence of the instability modes of the Tollmien-Schlichting type. Due to their amplification, the Tollmien-Schlichting waves eventually trigger the laminar-turbulent transition. This approach is also used by Pirozzoli and Grasso^[9]. The turbulence (see Fig. 1) appears downstream of the transitional region when x is approximately 220, and there is a sufficiently long distance for the development before interacting with the incident shock wave.

Simultaneously, we introduce an oblique shock with the angle β=33.2° from the upper boundary of the computational domain. The physical quantities ahead of the shock are related to those behind the shock by the Rankine-Hugoniot jump condition. The incident shock wave impinges on the turbulent boundary layer, and generates a reflected shock wave.

Through resolution study, the computational domain and the grid point number are finally chosen as those listed in Table 1. The grid points in the streamwise direction are uniformly located except in the shock-turbulent-interacting region (x∈[390, 450]), where the mesh is clustered due to the high resolution requirement. In the wall-normal direction, the grid points are clustered in the near-wall region according to a hyperbolic sine transformation, and the uniform grid points are located in the spanwise direction.

In the full-developed turbulent region, the grid spacing, expressed in terms of the viscous wall unit, is

where

In the above equation, υ_w is the kinematic viscosity at the wall, and u_τ is the friction velocity defined by

(1)

Throughout the paper, the subscript "w" represents the mean quantities at the wall, and the bar and the tilde over the quantities represent the Reynolds and Favre averaging, respectively. The minimum grid spacing in the streamwise direction occurs in the shock-turbulent-interacting region, where Δx⁺=3.03.

Because the simulation starts from the laminar boundary layer, the Blasius similarity solution is introduced as the inflow profile. The no-slip and non-penetration boundary conditions are imposed at the wall, except at the blowing and suction slot. The wall is assumed to be isothermal, and the wall temperature is approximately equal to the adiabatic wall temperature of the turbulent boundary layer, i.e., T_w=332 K. The non-reflecting boundary condition is used at the upper boundary. In order to diminish the perturbation-reflection from the outflow boundary, an additional sponge region is used from x=520 to x=575.6 with 120 grids stretching towards the downstream direction. The periodic boundary condition is used in the spanwise direction.

3 Baseline validation

In order to compare our simulation results with the reference data, we first compare the main properties of the turbulent profile at x=380 with those from Pirozzoli and Bernardini^[14] and Morgan et al.^[17] (see Table 2). Among the symbols in the table, ϕ represents the flow deflection angle, i.e.,

(2)

Re_δ, Re_θ, and Re_δ^∗ are the local Reynolds numbers whose reference lengths are the the nominal thickness (δ₀), the momentum thickness (θ), and the displacement thickness (δ^∗) of the boundary layer, respectively, H = δ^∗/θ denotes the shape factor, and the skin-friction coefficient is defined by

where τ_w is the shear stress at the wall defined by

The three oncoming Mach numbers in Table 2 are fairly close, but the Reynolds numbers of our simulation are apparently larger than those of the reference papers. It is satisfactory that quite good agreement on the sensitive quantities such as the shape factor and friction velocity is observed, and the slightly smaller value of C_f in our simulation is anticipated due to the larger local Reynolds number.

The mean streamwise velocity at the reference location x=380 is plotted in Fig. 2. Two empirical curves are shown in Fig. 2, which, respectively, represent the linear law

(3)

in the viscous sublayer and the logarithmic law

(4)

with κ=0.4 and B=5.5 in the mesolayer. The velocity and the wall-normal coordinate in Eq. (4) are normalized by wall units, i.e.,

(5)

where

(6)

is the van Driest transformation for compressible flows. The numerical mean-velocity agrees well with the two empirical curves in the regions y⁺ ≤ 5 and 30 ≤ y⁺≤100, respectively. The implication is that the fully-developed turbulent boundary layer has already been formed at this location. It has to be noticed that an obvious discontinuity is observed at y⁺ ≈ 1 000, which is induced by the incident shock wave.

The distribution of the skin-friction coefficient C_f and the wall pressure p_w in the vicinity of the interaction region are plotted in Fig. 3, where the streamwise coordinate is rescaled by (x-x₀) /L_sep as suggested by Morgan et al.^[17], in which x₀ = 436.69 denotes the nominal shock impingement point, and the separation length L_sep is defined as the distance between the separation point (x_sep=414.16) and the reattachment point (x_rea=428.7). From Fig. 3, we can see that our results agree well with those of Refs. [14] and [17]. The only difference is that the C_f curve in our simulation is lower than the reference curves outside the interacting zone, which is attributed to the relatively larger Reynolds number that we have selected.

Under the normalization suggested by Pirozzoli and Bernardini^[14], Fig. 4 exhibits the contours of the averaging and fluctuation velocities near the interaction region, where the subscript "rms" denotes the root-mean-square. In the sub-figures, the streamwise and wall-normal coordinates are normalized as follows:

(7)

respectively, where the interaction length L is defined as the distance between x₀ and the origin point of the reflected shock (x_on=402). From the figures, we can see that our results agree with those of Fig. 3 in Ref. [14] and Fig. 12 in Ref. [17].

Figure 5 shows the distribution of root-mean-squire of the wall pressure, the streamwise velocity, and the Reynolds stress. Although the overall agreement between our results and those of Ref. [14] is satisfactory, yet not surprisingly, there remain some nuances in the detail.

(ⅰ) A slightly higher wall pressure fluctuation arises at both the upstream and the downstream limits, which is understandable since the Reynolds number chosen in our simulation is larger. (ⅱ) The curves from our simulation oscillates more apparently, which is believed to be related to the insufficiency of the statistical time interval. Because the frequency of the shock oscillation is extremely low, it is too costly to include the shock motion in our sampling series. However, the time interval is much larger than the large-scale coherent structures in the turbulent flow. As pointed out by Piponniau et al.^[12], this could at most lead to an error of 1% on the first-and second-order statistics. This is why our results agree with those of Pirozzoli and Bernardini^[14] quantitatively.

4 Statistical properties 4.1 Mean flow

Figure 6 plots the distribution of the skin-friction coefficient (C_f). At the sufficient upstream locations, the values of C_f stay almost unchanged, drop dramatically in the early stage of the interaction region, and become negative after about x=414.16, which indicates the separation point. After attaining the minimum value at about x=416, the values of C_f vary gradually, until they reach the close neighborhood at the reattachment point at about x= 429. Then, the C_f curve increases with x monotonically. Simultaneously, as shown in Fig. 7, an APG is produced in the separation zone. The pressure starts to rise at the same dropping location of C_f. After peaking at about x=417.9, the pressure gradient gradually decreases when x approaches further downstream. However, its value has not reached zero up to the outflow boundary of the computational domain, implying that the flow field has not yet recovered to the upstream turbulent state.

The distribution of the heat flux q_w is plotted in Fig. 8, where

and k is the thermal conductivity. In the upstream undisturbed zone, because the wall temperature T_w is close to the turbulent recovery temperature, the wall heat flux is quite small. However, in the vicinity of the interaction zone, a large amount of heat is transferred to the wall and the q_w peaks inside the separation zone.

In order to monitor the detailed flow feature, 14 sampling points are selected, whose streamwise coordinates and their particular significance are summarized in Table 3.

Figures 9(a) and 9(b) plot the profiles of the mean streamwise velocity and temperature, respectively. The incident and reflected shocks in the inviscid region are apparent from the profile saltation. In the interacting region, where remarkable APG exists, some certain deficiencies appear on both the velocity profiles and the temperature profiles. With the decrease in the APG, the profiles gradually recover to the fully-developed turbulent profile. In the simulation, the separation zone is so tiny that it is inconspicuous in Fig. 9(a). We therefore zoom in the the mean-velocity profiles in the close neighborhood of the wall, i.e., 0≤ y≤ 0.1 (see Fig. 9(c)). Two separation bubbles with different sizes are clearly observed from the reverse-velocity zones, i.e., the larger one appears from s4 to s8, and the smaller one appears in the vicinity of s10. The two bubbles corresponds to the two valleys of the C_f curve (see Fig. 6).

4.2 Reverse velocity in the separation zone

A clearer observation on the separation bubbles are shown in Fig. 10(a), where the contours of the mean-velocity and the streamlines in the near-wall region are plotted.

As pointed out by Zheltovodov^[29-30], the reverse flow ahead of the forward-facing step can be considered as a near-wall jet, and the laminar and the turbulent profiles satisfy different distributions. Based on the suggestion, Loginov et al.^[20] demonstrated that there was a relaminarization effect in the separation bubble induced by a supersonic flow past a compression ramp. For the present simulation, we perform the same analysis on the mean-velocity profiles in the separation zone (see Fig. 10(b)). In the figure, the mean-velocity is normalized by the maximum value of the reverse flow u_max, and the wall-normal coordinate is normalized by the location ζ_1/2 at which the mean-velocity reaches half of its maximum value. The red solid and dashed lines in Fig. 10(b), respectively, represent the laminar and turbulent near-wall-jet profiles suggested by Vulis and Kashkarov^[31]. The velocity profiles from our simulation scatter around the laminar-jet profile, and keep far away from the turbulent profile, which implies that the flow in the separation zone is very close to the laminar state. No reverse transition from turbulent to laminar is observed for the oblique shock wave impingement on the turbulent boundary layers, which is different from the cases of the supersonic flow past compression ramps in Ref. [20].

Furthermore, in the vicinity of the separation zone, the APG induced by the oblique shock wave balances the total shear stress to the leading-order accuracy, i.e.,

(8)

where the total shear stress τ is the sum of the viscous stress and the Reynolds shear stress, i.e.,

(9)

Figure 11 plots the wall-normal distributions of both the viscous stress and the Reynolds stress in the near-wall region of the separated zone. The conspicuously common feature in the separation zone is that the viscous stress is dominant whereas the Reynolds stress is almost 0. In comparison with the behavior of the turbulent boundary layer with ZPG, the shear rate of the viscous stress is no more zero, but becomes nearly a constant for each x. As determined by Eq. (8), the shear rate balances the APG. Therefore, we compare the two quantities at different x (see Table 4). Both the shear rates of the viscous stress and the pressure gradient peak are at the same location (s6), and quantitative agreement is observed for all the locations.

4.3 Near-wall behavior of the mean-velocity in the interaction region 4.3.1 Introducing the pressure-velocity scale

For turbulent boundary layers with zero or small pressure gradient, the wall friction velocity u_τ, which is defined in Eq. (1), is selected as the reference velocity to demonstrate the near-wall behavior of the mean-velocity. Such a behavior is also described in Fig. 2 of the current paper. It is found that the linear law (3) and the logarithmic law (4) emerge in the viscous and meso sublayers, respectively.

However, the presence of the APG in the SWTBLIs breaks the balance of the viscous stress and the Reynolds stress in the turbulent boundary layer. Therefore, the traditional laws of the wall cease to be valid. Furthermore, in the reverse flow becomes negative, which leads to the invalidity of the definition (1).

For the incompressible turbulent boundary layer with the APG, Skote and Henningson^[23] suggested to choose the pressure velocity scale

(10)

as the reference velocity for normalization, and redefined the dimensionless velocity

(11)

and the dimensionless wall-normal coordinate

(12)

Figure 12(a) shows the distributions of u_τ and u_p in the neighbourhood of the interaction region, where the definition of u_τ is changed to

(3)

In the upstream of the separation zone, the value of u_τ is about 0.047, which is much larger than the value of u_p (about 0.003). In the vicinity of the separation point, u_τ falls rapidly, and is overwhelmed by u_p. In the separation zone, u_τ and u_p become comparable until they are in the close neighborhood of the reattachment point. After that u_τ recovers to the dominant factor.

In order to better compare the magnitudes of u_τ and u_p, we introduce

(14)

to denote their ratio. Figure 12(b) exhibits the distribution of γ near the interacting region. The ratio is much larger than 1 outside the separation zone. However, the ratio is of the order 1 inside the separation zone, and is close to zero near the separation and reattachment points.

4.3.2 Mean velocity in the near-wall region

Since the leading-order terms in the x-momentum equation are the pressure gradient, the viscous shear stress, and the Reynolds shear stress, Eqs. (8) and (9) can be recast to

(15)

Here, we assume that, p is a function of x, and , and ρ are functions of y since the quantities vary slowly along the other direction. Multiplying Eq. (15) by υ_w/(ρ_wu_p³), applying Eqs. (10) -(12), and introducing

give

(16)

Inserting Eqs. (11) and (12) into Eq. (13) yields

(17)

Integrate Eq. (16) with respect to y^p, apply the condition (17), and notice D(0) = W (0) = 1. Then, we obtain

(18)

where signs + and -represent the profiles without and with separation, respectively. It is noticed that the left-hand side of Eq. (18) is essentially the total shear stress.

4.3.3 Viscous sublayer

In the region very close to the wall, i.e., y^p = O(1), the Reynolds stress is negligible, i.e.,

Therefore, the dominant shear stress is the viscous stress, and this region is referred to as the viscous sublayer.

Integrate Eq. (18) with respect to y^p, and notice (D, W, u^p)→ (1, 1, 0) when y → 0. Then, we can obtain an explicit relation between u^p and y^p, i.e.,

(19)

Sufficiently far from the separation zone, where γ ≫ 1, Eq. (19) recovers to the linear law of the turbulent boundary layer with ZPG, i.e.,

(20)

Moreover, in the close neighborhood of the separation and the reattachment points, where γ ≪ 1, Eq. (19) becomes a parabolic law, i.e.,

(21)

in which the pressure gradient plays the leading role.

4.3.4 Mesolayer

The Reynolds stress comes to the leading order in the mesolayer, where y^p ≫ 1. We herein introduce a new characteristic velocity scale u_∗, which is defined by

(22)

where the signs + and -have the same definitions as those in Eq. (18). Multiplying Eq. (18) by u_p²/u_∗² gives

(23)

Based on the Boussinesq hypothesis, the Reynolds stress can be expressed as follows:

(24)

where υ_T is the eddy viscosity. According to the concept of the mixing length,

(25)

where κ=0.4 is the Karman constant (see Ref. [21]). Applying Eqs. (24) and (25) to Eq. (23) and noticing y^p≫ 1, we have

(26)

It can be seen that the viscous stress plays a minor role in the mesolayer.

For compressible flows, the van Driest transformation is introduced, i.e.,

(27)

Then, the solution of Eqs. (26) is obtained to be

(28)

for the profiles without separation and

(29)

for the profiles with separation, where C is constant.

If γ ≫ 1, where the pressure gradient is very small, Eq. (28) recovers to the logarithmic law

(30)

where B=5.5 is the constant from the classical turbulent theory. In the vicinity of the separation and the reattachment points, where γ ≪ 1, both Eqs. (28) and (29) satisfy a squire-root law

(31)

implying a natural communication when the profiles cross the zero-shear points.

4.3.5 Verification by the DNS data

Figure 13 shows the comparisons of the near-wall mean-velocity between the DNS data and the above analysis, where the dashed lines are from Eq. (19) for the viscous sublayer and Eq. (28) or Eq. (29) for the mesolayer, while the dot-dashed lines are from Eq. (20) for the viscous sublayer and Eq. (30) for the mesolayer. The latter are essentially the same as the linear and logarithmic laws (3) and (4) of the classical turbulent theory for the ZPG.

At all the streamwise locations, the present laws of the wall can predicate the mean-velocity profiles to high accuracy in both the viscous sublayer and the mesolayer. However, the traditional near-wall laws show poor agreement with the DNS data, especially inside the separation zone, which indicates the significant role played by the pressure gradient.

Admittedly, the constant C in Eqs. (28) and (29) are derived from curve fitting instead of from analytical approach. The reason is that the matching condition from the outer region of the boundary layer is not determined due to the effect of the impinging shock wave. Nevertheless, the importance of the new law of the wall is clear because the dominant mechanism is obtained.

It is also noted that the buffer layer connecting the viscous sublayer and mesolayer thickens inside the separation zone. We however have not derived the explicit relation for this layer due to its complexity.

For the turbulent boundary layer with the ZPG, the second-order statistics in the near-wall region, such as the Reynolds stress, also exhibit universal relations (local Reynolds number independent) when plotting against y⁺. However, for the SWTBLI problem, such curves at different streamwise locations differ from each other remarkably (see Fig. 14(a)).

Gungor et al.^[24] suggested that the characteristic stress scale should be ρ_wu_p² instead of ρ_wu_τ² for the turbulent boundary layer with the APG. Noticing that the APG only concentrates in the close neighborhood of the separation zone for the current problem, we therefore choose ρ_wu_∗² as the reference stress scale. Thus, we introduce

(32)

(33)

and plot -〈ρu''v''〉^∗ against y^∗ for different streamwise locations (see Fig. 14(b)). All the curves exhibit the same trend both qualitatively and quantitatively.

5 Conclusions

The interaction of an impinging oblique shock wave with a supersonic turbulent boundary layer at the Mach number 2.25 is investigated by the DNS. The main purpose of this paper is to provide a comprehensive analysis on the near-wall behaviors of the interaction, such that further works on improving the accuracy of the turbulent modeling could be inspired.

Our numerical results are compared with the reference data from Pirozzoli and Bernardini^[14] and Morgan et al.^[17] with similar configurations, and are shown to be of high accuracy and reliable. Additional efforts have been made on providing the detailed turbulent statistics such as the distribution of the surface friction coefficient, the pressure gradient, the heat flux, and the mean flow profiles. These results are adequate to serve as a standard data-set on further turbulence modeling validations.

The reverse flow inside the separation zone is found to be laminar through the analysis of the local mean profiles, which implies that the Reynolds stress here is negligible. Simultaneously, the viscous stress linearly increases with the wall-normal coordinate y, instead of being a constant for the ZPG cases. The shear rate of the viscous stress agrees perfectly with the local pressure gradient, indicating the balance between the viscosity and the pressure gradient in this region. Therefore, a new relation in the viscous sublayer is formed due to the additionally dominate role of the pressure gradient, which, being different from the traditional linear law for the ZPG, exhibits a parabolic dependence (19) (with a negative sign in front of γ) of the mean-velocity on y. Right above this sublayer, the Reynolds stress comes to the leading order as well. Therefore, the leading balance is among the viscosity, the pressure gradient, and the Reynolds stress. This region is referred to as the buffer layer, and no explicit relation has been obtained so far. As y further increases to the mesolayer, the viscosity becomes negligible, and the balance between the pressure gradient and the Reynolds stress determines the relation (29).

In the regions upstream and downstream of the separation zone, the parabolic law (19) (with a positive sign in front of γ) of the viscous sublayer still holds, and it gradually recovers to the linear law (20) when the pressure gradient reduces successively. Moreover, the relation in the mesolayer is expressed as Eq. (28) in the attached region, and also recovers to the logarithmic law (30) in the small pressure-gradient limit. The two relations (29) and (28) in the mesolayer communicate with each other at the separation and reattachment points for the common squire-root dependent of on y.

The new laws of the wall proposed in this paper are proved to be effective on reproducing the mean-velocity profiles from the DNS results both inside and outside the separation zone. The distribution of the Reynolds stress in the near-wall region is remarkably regularized under the normalization approach utilized in the new law of the wall.

Acknowledgements The authors are grateful to Professor Xinliang LI of Institute of Mechanics, Chinese Academy of Sciences for kindly providing the DNS code. Professors Heng ZHOU and Jisheng LUO of Tianjin University are also acknowledged for their valuable discussion and suggestions. This work is carried out at the National Supercomputer Center in Tianjin, and the calculations are performed on TianHe-1 (A).