Inflow boundary condition for DNS of turbulent boundary layers on supersonic blunt cones

Transcription

1 ppl. Math. Mech. -Engl. Ed., 8, 9(8): DOI 1.17/s c Shanghai University and Springer-Verlag 8 pplied Mathematics and Mechanics (English Edition) Inflow boundary condition for DNS of turbulent boundary layers on supersonic blunt cones DONG Ming ( ) 1,, ZHOU Heng ( ) 1 (1. Department of Mechanics, Tianjin University, Tianjin 37, P. R. China;. Tianjin Key Laboratory of Modern Engineering Mechanics, Tianjin 37, P. R. China) (Contributed by ZHOU Heng) bstract For direct numerical simulation (DNS) of turbulent boundary layers, generation of an appropriate inflow condition needs to be considered. This paper proposes a method, with which the inflow condition for spatial-mode DNS of turbulent boundary layers on supersonic blunt cones with different Mach numbers, Reynolds numbers and wall temperature conditions can be generated. This is based only on a given instant flow field obtained by a temporal-mode DNS of a turbulent boundary layer on a flat plate. Effectiveness of the method is shown in three typical eamples by comparing the results with those obtained by other methods. Key words turbulent boundary layer, direct numerical simulation, supersonic, blunt cone, inflow condition Chinese Library Classification O , O543.3 Mathematics Subject Classification 76F65 Introduction With the development of the aeronautics and aerospace technology in recent years, problems regarding the compressible turbulent boundary layer on cones have been attracting more and more attention. Due to the fact that eperiments can not provide detailed information about the flow field, and since it may cost too much in case of supersonic flows, the DNS method plays a more and more important role in turbulence research, especially for supersonic flows. Dong and Luo [1 3] have performed spatial-mode DNS for the transition and turbulence of supersonic and hypersonic boundary layers on sharp and blunt cones. Pirozzoli et al. [4] and Li Xinliang et al. [5] also performed similar computations for turbulent boundary layers on flat plates. The computational methods they used (which hereafter will be called normal DNS method) were all the same, namely, starting from basic laminar solutions, and by introducing certain finite amplitude disturbances at the inlet of the computational domain, transition was triggered at a certain downstream location. Such method corresponds to the physical processes of natural transition. However, the computational domain in the stream-wise direction has to be long enough to allow for the transition process to be included, resulting in high computational costs. Received Jun. 16, 8 / Revised Jun. 4, 8 Project supported by the National Natural Science Foundation of China (Nos. 1635, 97167) and the Special Foundation for the uthors of National Ecellent Doctoral Dissertations (No. 38) Corresponding author ZHOU Heng, Professor, hzhou1@tju.edu.cn

2 986 DONG Ming and ZHOU heng If research is only concerned with the turbulent stage, the inclusion of transition is unnecessary. Especially, when the study covers turbulence with many different parameters, the computational cost may be unacceptable if the transition process is included. In addition, without the breakdown process of transition, this may sometimes cause the computation divergence, and make the computation more unstable. In the normal DNS method, there are many ways to set the inflow conditions, because the inlet of the computational domain is laminar flow. However, if the computation begins with turbulent flow, the generation of an appropriate inflow condition is a difficult problem, not properly solved yet. ased on the statistical theory of turbulence eistence, Lund et al. [6] proposed a method to generate the inflow condition for a spatial-mode DNS of a turbulent boundary layer on a flat plate. Through numerical verification, it was found that the transient stage due to the imperfect inflow condition is short. However, the method is only applicable for the incompressible turbulent boundary layer on a flat plate, not applicable for compressible flows, whose statistical theory is not yet well founded. In the case of a turbulent boundary layer on a flat plate with an oncoming Mach number 4.5, Huang and Zhou [7] proposed a new method to choose the inflow condition for spatial-mode DNS. Firstly, the temporal-mode DNS of a supersonic turbulent boundary layer on a flat plate was performed, and then using only one instant flow field of fully developed turbulence, the inflow condition for a spatial-mode DNS with the same Mach number, Reynolds number and wall condition was deduced. The fully developed stage of turbulence was easily reached with only a very short transient stage, too. Furthermore, they showed that the inflow condition for the spatial-mode DNS with a different Reynolds number, but with the same Mach number and wall condition, could be also be generated by using the same flow field. ecause the computational cost of a temporal mode is much smaller than that of a spatial mode DNS, the proposed method is very tempting. However, for the boundary layer problem on a cone, the periodic condition in stream-wise direction is not applicable, thus we cannot even perform a temporal-mode DNS. Whether the above method can be etended to the case of the boundary layer on a cone is far from obvious. In deducing the inflow condition for the spatial-mode DNS from the flow field obtained by temporal-mode DNS, a time-space transformation is needed, in which there is a velocity term, and what Huang and Zhou used was the mean propagation velocity of coherent structures in the wall region of the turbulent boundary layer. The implication is that if the characteristics of coherent structures of turbulent flow at the inlet are properly reproduced, the inflow condition should be proper. If the characteristics of the coherent structures in turbulent boundary layers are really universal, then we can even deduce the inflow condition for the spatial-mode DNS of the turbulent boundary layer on a cone from an instant flow field obtained from a temporalmode DNS for a boundary layer on a flat plate. We can also epect that, the inflow condition for flows with different Mach numbers, Reynolds numbers and wall temperature conditions should be deductible from the same flow field data. The instant turbulent flow field from temporal-mode DNS contains the following information: the mean flow and the instant fluctuation flow field. proper inflow condition should satisfy the following conditions: (1) the mean flow at the inlet should be close to the actual mean flow, and () the fluctuation at the inlet should be close to the actual fluctuation flow. In deducing the inflow condition of a spatial-mode DNS, the first one is easily satisfied. However, there are many different scales for the fluctuation, and their propagation speeds are, in general, not the same. It is inappropriate to use the same time-space transformation for their propagation, because it does not reflect their different entering speeds through the inlet, yet obviously, it is also impractical to choose different time-space transformations for different scales of fluctuations. The solution is to use the propagation speed of the main fluctuation, which plays the most important role in the generation of the turbulence, in the time-space transformation. ased on our present knowledge, the fluctuation corresponding to coherent structures should be the

3 Inflow boundary condition for DNS of turbulent boundary layers 987 main fluctuation. This was the reason why Huang and Zhou [7] used the mean propagation speed of coherent structures in their time-space transformation, and the result turned out to be satisfactory. In the following, the flow field at a certain instant, obtained by Huang and Zhou [7] in their temporal-mode DNS of a turbulent boundary layer on a flat plate with Mach number 4.5, will be used to deduce the inflow conditions for our spatial-mode DNS of turbulent boundary layers on a supersonic cone. s shown in Fig. 1, firstly, the chosen flow field is stretched in a normal-wise direction to make its boundary layer thickness equal to the boundary layer thickness at the inlet of our computational domain for the turbulent boundary layer on a cone (which makes the Reynolds numbers based on the local boundary layer thicknesses the same in both cases). Secondly, the solution is transformed from rectangular coordinates into the body-fitted coordinates used for our cone. Then, the flow field is divided into two parts: the mean flow and the fluctuation. The mean velocity profile can be directly used at the inlet of the computational domain for the spatial-mode DNS, however, the mean temperature and mean density should be modified according to the oncoming Mach number and the wall condition. The fluctuation part is put at the inlet, under a certain time-space transformation, together with the mean flow. ecause there is no fluctuation outside of the boundary layer, the flow field there can be set as the laminar solution or the inviscid solution. The detailed method will be further clarified in Section 1. Fig. 1 Sketch of the inflow condition In this paper, results of three test cases will be given, namely, DNS results of turbulent boundary layers on a supersonic blunt cone with semi-angle 5, oncoming Mach number 4.5,.5 and 6. The results will show that the proposed method is workable. 1 Numerical method The numerical method used in this paper is the same as that in Ref. [], ecept for the inflow boundary condition. The coordinate is a body-fitted coordinate (, y, θ), where is the coordinate along a generatri of the cone, starting from the ape of a sharp cone, corresponding to our blunt cone, y is a coordinate in the normal-wise direction, they are all non-dimensionalized by the radius R of the cone nose, θ is the angular coordinates in the circumferential direction. ll the physical quantities are non-dimensionalized by their counterparts in the uniform oncoming flow out of the shock. The details for deducing the inflow condition are given as follows.

4 988 DONG Ming and ZHOU heng 1.1 The normal-wise stretching of the temporal-mode turbulent flow field and the matching of the span-wise coordinates Firstly, we need to stretch the known temporal-mode turbulent flow field on a flat plate in the normal-wise direction, to make its boundary layer thickness equal to that of the spatialmode turbulent boundary layer at the inlet on our blunt cone, and the latter thickness needs to be estimated from the laminar solution. lso, we have to transform the flow field under rectangular coordinates to a flow field under the body-fitted coordinates used for the blunt cone. For convenience, in the following, variable X is used to represent physical quantities in the flow field, such as density ρ, velocities u, v, w in the stream-wise, normal-wise and span-wise directions, temperature T, and pressure p. Let ˆX T (ξ, η, ζ) be the flow variable of the temporal-mode turbulent flow field at a certain instant, where ξ, η, ζ are the stream-wise, normal-wise and span-wise coordinates, respectively, under rectangular coordinates; and ˆX T (ξ, η, ζ) is non-dimensionalized by its value at the outer edge of the boundary layer. It can be decomposed into an averaged value and the fluctuation, such that ˆX T (ξ, η, ζ) = ˆX T (η) + ˆX T (ξ, η, ζ), where an over-bar implies the average with respect to ξ and ζ, and a prime represents the fluctuation. We can find the nominal thickness of the boundary layer, denoted as ˆδ T. The laminar solutions of the boundary layers on the blunt cone can be found by using the method proposed in Ref. []. The laminar solution at the inlet of the computational domain selected is denoted by X c (y), and its corresponding nominal boundary layer thickness is denoted as δ c. Suppose the flow just finished its transition stage right before the inlet of the computational domain, then it is already turbulent at the inlet. Let the physical quantities at the inlet of the computational domain for the spatial-mode DNS of the turbulent boundary layer on the blunt cone be denoted by X T (y, θ, t); they can also be written as the sum of its averaged value, with respect to θ and t, and the fluctuation, such that X T (y, θ, t) = X T (y) + X T (y, θ, t), and the corresponding nominal boundary layer thickness is denoted as δ T. We can estimate δ T from the abovementioned δ c. Let δ T = k δ δ c. (1) In his study on the transition of the incompressible boundary layer on a flat plate, Tang Hongtao found that [8] the nominal thickness of the turbulent boundary layer at the ending of transition is usually to 3.5 times that of the laminar boundary layer at the beginning of the transition. We can thus choose the value of k δ within this range. The turbulent profile ˆX T obtained by temporal-mode DNS can be stretched in the η direction according to the similarity of turbulence, to make the nominal boundary layer thickness equal to δ T. Moreover, we need to transform the solution of turbulent data under rectangular coordinates to body-fitted coordinates used for the cone. The simplest way is to let their normal coordinates correspond to each other directly, i.e., let η = y; and the span-wise coordinate ζ corresponds directly to the circumferential radian coordinate on the surface of the blunt cone, under the principle that the unit arc length on the cone surface is equal to the unit length of the flat plate, such that ζ = ( sin α) θ, where is the stream wise coordinate at the inlet of the computational domain on blunt cones, and α is the cone semi-angle. One has to notice that in the computation for the blunt cone boundary layer, the flow variables are always non-dimensionalized by their corresponding quantities in the uniform oncoming flow out of the shocks, so their non-dimensional values at the outer edge of the boundary layers are not equal to 1. However, for boundary layers on flat plates, the reference quantities for nondimensionalization, no matter if chosen from their values in the uniform oncoming flow or their values at the outer edge of the boundary layer, are the same, so the non-dimensional value of each flow variable is always equal to 1 at the outer edge of the boundary layers. Therefore, in

5 Inflow boundary condition for DNS of turbulent boundary layers 989 transforming ˆX T into the inflow condition, we need to proportionally amplify the values of ˆX T first. That is, to let X T (ξ, y, θ) = k u ˆXT (ξ, ˆδ T /δ T y, sin α θ), () where k u is the ratio of the laminar solution X c on the blunt cone to the turbulent solution ˆX T on a flat plate; both take their values at the outer edge of the boundary layer. That is to say, for ρ T, ŭ T, v T, w T, T T, p T, k u should be ρ c (δ T )/ˆρ T (ˆδ T ), u c (δ T )/û T (ˆδ T ), v c (δ T )/û T (ˆδ T ), u c (δ T )/û T (ˆδ T ), T c (δ T )/ ˆT T (ˆδ T ), p c (δ T )/ˆp T (ˆδ T ) respectively. fter this modification, the variable of the turbulent flow field can again be written as the sum of its averaged value with respect to ξ and θ, and the fluctuation as: XT (ξ, y, θ) = X T (y) + X T (ξ, y, θ). 1. The selection of the mean flow at the inlet of the computational domain The reason why the flow variables at the inlet are decomposed into its mean value and the fluctuation is that we epect the fluctuation, especially that induced by coherent structures, to have certain universal characteristics. On the other hand, although the mean quantities are certainly not universal, they can be easily modified to fit the practical situation, as shown in the following paragraphs. No matter if it is for a cone or flat plate, if the Mach number is high or low, if the boundary condition at the wall is isothermal or adiabatic, the Van Driest transformed mean streamwise velocity distribution always satisfies the near-wall law of incompressible flows in the wall region [ 4,9 1], while in the outer region of the boundary layer, it always asymptotically approaches the value outside the boundary layer. That is to say, the distribution of the nondimensional mean velocity is more or less universal. Therefore, the non-dimensional mean velocities u T (y), v T (y), w T (y) within the boundary layer can be given directly by the modification solution (ŭ T (y), v T (y), w T (y)) obtained in Section 1.1. The above solution within the boundary layer is made to join smoothly with the laminar solutions (u c (y), v c (y), w c (y)) outside of the boundary layer. So eventually we let u T = ŭ T (1 f((y y 1 )/(y y 1 ))) + u c f((y y 1 )/(y y 1 )), v T = v T (1 f((y y 1 )/(y y 1 ))) + v c f((y y 1 )/(y y 1 )), (3) w T = w T (1 f((y y 1 )/(y y 1 ))) + w c f((y y 1 )/(y y 1 )), (y ), where the transitional function f(y) is defined as f(y) = (tanh(7y 4) + 1)/ ( < y < 1), 1 (y 1); y 1, y are two points outside the boundary layer, and y 1 < y. The formula (3) implies that in the range y = y 1, the mean velocity is given by the modified turbulent profile X T ; in the range of y > y, it is given by the laminar solution X c of the boundary layer on the blunt cone, and the range y 1 y y is transitional. If the Mach number and wall temperature condition for the case of a blunt cone are the same as those for the case of flat plate, and the result from the temporal mode DNS of the latter is what we would use to deduce the inflow condition, then the mean temperature and the mean density can be written in the same form as in equation (3), that is, { T T = T T (1 f((y y 1 )/(y y 1 ))) + T c f((y y 1 )/(y y 1 )), (4) ρ T = ρ T (1 f((y y 1 )/(y y 1 ))) + ρ c f((y y 1 )/(y y 1 )). However, if the Mach numbers or wall temperature conditions are not the same, there will be big differences between the mean thermodynamic variables T T, ρ T and T T, ρ T, so the above method can no longer be used, and one should take the wall temperature into consideration to find approimate distributions for the mean temperature and mean density, which are as close as possible to the real situation.

6 99 DONG Ming and ZHOU heng For the turbulent boundary layer on a flat plate, White [11] proposed a formula which links the mean temperature uniquely with the mean velocity. The formula has been verified many times by DNS results [4,1,1 13]. However, for turbulent boundary layers on blunt cones, the results of mean temperature calculated by the formula are lower than the DNS results [ 3]. Considering that for an adiabatic wall, the wall temperature would be close to the total temperature of the oncoming flow, we suppose T aw = T e + u e /C p, where the subscript e implies the value at the outer edge of the boundary layer, and C p is the specific heat at constant pressure. Then, according to White s formula, the mean temperature can be given as follows: T T = T w + (T aw T w)u T /u e u T /C p, (5) where T w is the wall temperature, and for an adiabatic wall T w = T aw. The mean density should satisfy the equation of state: ρ T = (ρ T /R ρ T T T )/T T, where R is the gas constant. nd from the turbulent boundary layer equation (p+ρv ) y =, we can deduce p T p c ρ T v T, so we have ρ T (p c /R ρ T v T /R ρ T T T )/T T. Or we can simplify further as ρ T p c /(T T R). (6) In turbulent boundary layers, the transport caused by turbulent fluctuations plays the leading role. If the fluctuating part of the inflow is correctly given, the turbulence would reach its fully developed state quickly, while a certain amount of inaccuracies in the given mean flow quantities may not have a big effect. 1.3 The selection of the fluctuating part of the inflow at the inlet of the computational domain In order to transform the instantaneous fluctuation flow field, obtained by the temporal-mode DNS, into the inflow condition at the inlet of the computational domain of the spatial-mode DNS, we need the propagation speed of the fluctuations. Knowing the propagation speed, the time-space transformation can readily be obtained. However, different scale fluctuations may have different propagation speeds, and one can hardly distinguish different scale fluctuations from each other. However the fluctuation caused by coherent structures in the wall region plays the main role in turbulence, so in the transformation, the mean propagation speed of coherent structures is a good candidate, thus we have t = ξ/c, (7) where c is the mean propagation speed of the coherent structures, and according to Ref. [7], it is taken to be.9. In addition, the value of the instantaneous pressure satisfies the equation of state: p T = ρ T RT T. Numerical results.1 Turbulent boundary layer on a blunt cone with oncoming Mach 4.5 and adiabatic wall Results from the temporal-mode DNS for a turbulent boundary layer on a flat plate performed by Huang and Zhou [7] are chosen to deduce the inflow condition of our spatial-mode DNS for a turbulent boundary layer on a blunt cone. The parameters in Huang and Zhou s DNS were: the oncoming Mach number is 4.5, the wall temperature condition is adiabatic, the dimensions of the computational domain are L L y L z = , the number of meshes is N N y N z = , the width of the meshes is y w + z = , L, L y, L z, and z are non-dimensionized by the initial laminar boundary layer displacement thickness, while y w + is the normal-wise mesh size at the wall, measured in wall units.

7 Inflow boundary condition for DNS of turbulent boundary layers 991 Our blunt cone has a semi-angle of 5, the temperature of the oncoming flow is 79K, the wall condition is adiabatic, the Reynolds number based on the cone nose radius and the oncoming parameters is 1, and the location of the inlet of the computational domain corresponds to = In this section, three test cases, denoted as,, and C, will be studied. The widths of the meshes are y + w θ =.R.1.17, the normal-wise etent of the computational domain is 4R, the circumferential etent of the computational domain is 16.3, so at the inlet, the length of the arc on the cone surface, corresponding to this angle, matches the span-wise etent of the computational domain on the flat plate. However, the stream-wise numbers of the meshes for the three cases are not the same, nor are the initial thicknesses of the turbulent boundary layer at the inlet (that means the numbers of k δ are different). In this section, there are two ways to compute the mean temperature and the mean density: (1) Considering that the cone semi-angle is small, the shock in front of the blunt cone is weak ecept near the nose. Consequently, the Mach number at the outer edge of the boundary layer of the blunt cone is close to the oncoming Mach number (through computation, it is found that the Mach number at the outer edge of the boundary layer is 4.8 for a cone with semi-angle is 5, if the Mach number of the oncoming flow is 4.5), so we may assume that the flow variable distributions in the boundary layers of the blunt cone and flat plate are close to each other, thus equation (4) can be used to compute the mean temperature and mean density. () Use equations (5) and (6) to compute the mean temperature and mean density. In the following, we will compare results obtained by using both methods. The parameters for the three test cases are shown in Table 1. Table 1 Parameters for the three cases in test 1 Case k δ N N y N z Method to compute T and ρ Eq. (4) Eq. (4) C Eqs. (5),(6) The fluctuation of velocities and thermodynamic variables are given by methods introduced in Section 1.1, namely, we put {u T, v T, w T} = {ŭ T, v T, w T}, (8) {ρ T, T T } = { ρ T, T T }. (9) Shown in Fig. are the ( stream-wise ) distributions of the skin friction coefficients for the three u cases, in which C f = µ w y /ρ u, where the subscript w implies values at the wall, and w the subscript implies values for the oncoming flow; µ is the viscous coefficient, determined by the Sutherland formula. ecause the initial boundary layer thicknesses are different for the three cases, their mean velocity gradients at the wall are different near the inlet, and thus also for C f. However, further downstream, C f for all three cases come closer to each other, especially for cases and C. Figure 3 shows the stream-wise distributions of the boundary layer displacement thicknesses for the three cases, in which δ = δ (1 ρū/ρ eu e )dy, and δ is the nominal of the boundary layer thickness. s it can be seen, in the range of < 95, the evolution of the boundary layer thicknesses are not very smooth for the three cases, but further on, they become smoother and parallel to each other. The implication is, to reach a fully developed state of turbulence, a stream-wise length of 1R is needed for their adjustment. The value of k δ in equation (1) may influence the thickness of the turbulent boundary layer calculated, and the larger the value

8 99 DONG Ming and ZHOU heng of k δ is, the thicker the turbulent boundary layer will be. There may be two reasons making k δ different. Firstly, as the results of Tang [8] show, different initial disturbances triggering the transition may result in different initial turbulent boundary layer thickness. Secondly, it may be caused by different transition locations. ll the results can be seen as reasonable. The C f curves and boundary layer thicknesses are very close for cases and C, and the difference in these two cases lies in their differences in calculating the mean temperature and mean density. This implies that the use of equations (5) and (6) to compute the mean temperature and mean density is reasonable. Case C is selected to study the mean flow profiles at the fully developed stage of turbulence; the results of the other two cases are very close to that of case C, and so are not shown. The normal wise distributions of the Van Driest transformed stream-wise mean velocity at locations = 97 and 13 are shown in Fig. 4, where y + is in wall unit, u + c is the Van Driest transformed mean velocity, non-dimensionalized by skin friction velocity (for its definition, see Ref. [1]). The two curves both satisfy the linear law u + c = y+ and the log-law u + c =.5 lny in the region y + 8 and y +, respectively, and show very good agreement. They also agree well with results in Refs. [ 3], the only difference being the width of the log-law region. possible reason is that their oncoming Mach numbers are different. C f Fig. Skin friction coefficient C δ* C Fig. 3 Displacement thickness of boundary layer 5 u + c u + = y + u + =.5lny DNS- = 97 DNS- = y + Fig. 4 Van Driest transformed stream-wise mean velocity s shown in Refs. [ 3], the mean profiles of fully developed turbulent boundary layers on blunt cones bear similar natures. The normal-wise distributions of the Favre-averaged (for its definition, see in Ref. []) mean profiles at locations = 93, 98, 1, and 13 in case C, nondimensionalized by their values at the outer edge of the boundary layer, are shown in Fig. 5, where the mean stream-wise velocities are shown in Fig. 5(a), and the mean temperatures are shown in Fig. 5(b). The normal-wise coordinates are non-dimensionalized by the local displacement thickness of the boundary layers. s it can be seen, the mean profiles of velocity and temperature are similar starting from = 98, but are a little different from the profiles

9 Inflow boundary condition for DNS of turbulent boundary layers 993 at = 93. This is because the inflow condition is not accurate enough to match the fully developed turbulence, so a certain adjustment is needed, and the location = 93 is still within this stage. y/δ* 3 1 = 93 = 98 = 1 = 13 y/δ* 3 1 = 93 = 98 = 1 = u/u ~ ~ e (a) Mean velocity ~ ~ T/T e (b) Mean temperature Fig. 5 Mean velocity and temperature profiles at different locations of the turbulent boundary layer From the results shown in this section, the proposed inflow condition does work. In the following section, turbulent boundary layers with other Mach numbers and wall temperature conditions will be tested, and the results are compared with those obtained by the normal DNS method.. Turbulent boundary layer on a blunt cone with oncoming Mach.5 and an adiabatic wall In this case, the turbulent boundary layer on the same blunt cone with the same adiabatic wall conditions and the same temperature of the oncoming flow but with a different Mach number.5 and a different Reynolds number 5, is studied. The results are compared with results obtained by the normal DNS method in Ref. [3]. The inflow condition of Mach.5 is again deduced from the same flow field data used in the previous section. However, to deduce the inflow condition in the case of Mach.5 from the data for Mach 4.5, not only the mean quantities, but also the fluctuation quantities need to be modified. It is found in our computation that the fluctuation of thermodynamic variables calculated by equation (9) are of the same order of magnitude as the mean quantities in the near-wall region, which may cause the computation to be unstable. s shown in Refs. [] and [3], the higher the oncoming Mach number is, the bigger the root mean square (rms) values of the fluctuation of the thermodynamic variables will be. ssuming for the moment that the rms value of the fluctuation of the thermodynamic variables are in direct proportion to the Mach number, the fluctuation of density and temperature can be modified as follows: {ρ T, T T } = { ρ T, T T } (M/ ˆM), (1) where ˆM is the Mach number of the temporal-mode turbulence on a flat plate, and M is the oncoming Mach number for our blunt cone. The same modification is applied to velocity fluctuations, such that {u T, v T, w T} = {ŭ T, v T, w T} (M/ ˆM). (11) In this section, three test cases are studied, denoted as cases,, and C. The number of meshes is y + w θ =.36R.49 (.114,.1,.11 ), respectively, the normal wise etent of the computational domain is R for all cases, and the circumferential etents of the computational domain are 14.6, 13.1, 14.1, respectively. Notice that both equations (8) and (11) have been tested. The parameters for the three cases are shown in Table.

10 994 DONG Ming and ZHOU heng Table Parameters for the three cases in test Case Location of inlet k δ N N y N z Method to compute u, v, w = Eq. (8) = Eq. (8) C = Eq. (11) The stream-wise distributions of the skin friction coefficients are shown in Fig. 6, where the solid line is the result of Ref. [3], and the curves marked with, and C are the results of the three cases in this section. Similar to what we have seen in Section.1, there is always a region whose length is 1R near the inlet for the adjustment of C f curves; afterwards, the curves show good agreement with that computed by the normal method, confirming that the proposed method of deducing the inlet condition is workable, at least for the computation of the C f curve. The stream-wise distributions of the displacement thicknesses of the boundary layers are shown in Fig. 7, from which one can see that these curves are parallel to each other after their initial adjustment. The differences in the boundary layer thicknesses are due to either that they have different initial thicknesses, as the k δ values are not the same, or the inlet locations are not the same. The Van Driest transformed stream-wise mean velocity at the location = 1 88 for the three cases, together with that taken from Ref. [3], are shown in Fig. 8. One can see that the four curves show good agreement with each other, and in the ranges y + 8 and y + 15, they all satisfy the linear law u + c = y+ and log law u + c =.5 lny , respectively. C f.3..1 Ref.[3] C δ Ref.[3] C Fig. 6 Skin friction coefficient Fig. 7 Displacement thickness of the boundary layer u Ref.[3] C u + = y + u + =.5lny y + Fig. 8 The Van Driest transformed stream-wise mean velocity The normal-wise distributions of the mean velocity and mean temperature at locations = and 1 14 are shown in Fig. 9, including the results taken from Ref. [3] and the results for cases, and C. s one can see, the agreement is very good, and they do satisfy the similarity of having turbulent profiles.

11 Inflow boundary condition for DNS of turbulent boundary layers 995 y/δ* 6 4 Ref.[3]_1114 Ref.[3]_114 _1114 _114 _1114 _114 C_1114 C_114 y/δ* Ref.[3]_1114 Ref.[3]_114 _1114 _114 _1114 _114 C_1114 C_ u/u ~ ~ ~ ~ e T/T e (a) Stream-wise velocity (b) Temperature Fig. 9 Comparison of the mean profiles at the location = 1114 and The above results imply that the method for generating the inflow condition proposed in this paper does work, and both Eq. (8) and Eq. (11) can be used. However, it is obvious that as the oncoming Mach number increases or decreases indefinitely, it is surely impossible for the rms values of the fluctuation quantities to become either infinitely large or infinitely small, so some nonlinear terms should be included in Eq. (11), or at least one should notice that it can only be used when the Mach number is within a certain range..3 Turbulent boundary layer on a blunt cone with oncoming Mach 6 and an isothermal wall In this section, the turbulent boundary layer on the same cone as in Section.1, with the same temperature of oncoming flow and Reynolds number, but with Mach number 6 and an isothermal wall (whose wall temperature is 94K), will be studied. Results are compared with those obtained by the normal DNS method in Ref. []. Two cases are studied, and the computational conditions are: the normal-wise etent of the computation domain is R, the circumferential etent of the computation domain is (14.3, 15.7 ), and the mesh size is y + w θ =.4R Equation (8) is used to get velocity fluctuations. However, for an isothermal wall, the temperature fluctuation is zero at the wall. Considering that the value of the pressure fluctuation there is also small, the density fluctuation is close to zero there, too. We thus have to modify Eq. (1) near the wall, hence let {ρ T, T T } = { ρ T, T T } (M/ ˆM) f(y/(bδ T )), (1) where f(y) is the transitional function defined in Eq. (3), bδ T is the width of the transitional domain, b is taken as.5, and δ T is the nominal thickness of the turbulent boundary layer. Equation (1) is actually Eq. (1) with a certain near-wall modification. Equation (1) without Mach number modification is also used, such that {ρ T, T T } = { ρ T, T T } f(y/(bδ T)). (13) Since in this case, the density and temperature fluctuations will not be too large to cause the computation to fail. The parameters for the two cases are shown in Table 3. Table 3 Parameters for the two cases in test 3 Case Location of inlet k δ N N y N z Method to compute T and ρ = Eq. (1) = Eq. (13) The stream-wise distributions of the skin friction coefficients are shown in Fig. 1, where the result showing the solid line is taken from Ref. [], and the curves and are computed

12 996 DONG Ming and ZHOU heng by the method proposed in this paper. One can see again that near the inlet there is a certain adjustment due to the imperfect inflow conditions, but not far away, about 1R downstream, the curves show ( good) agreement. The stream-wise distributions of the heat transfer rate at T the wall q w = k w y, where k is the heat-transfer coefficient determined by the Sutherland w formula, are shown in Fig. 11. gain, after a certain adjustment, about 1R downstream from the inlet, the three curves show good agreement, too. The distributions of the boundary layer thickness are shown in Fig. 1, where the displacement thicknesses of the boundary layer are shown in Fig. 1(a), and the shape factor (the ratio between the displacement thickness and the momentum thickness) is shown in Fig. 1(b). s it can be seen in the fully developed turbulent stage, i.e., 1R downstream from the inlet, the displacement thicknesses computed by the method proposed in this paper are parallel to the results taken from Ref. [], and the shape factors show good agreement, too C f Ref.[] q w Ref.[] Fig. 1 Skin friction coefficient Fig. 11 Skin heat-transfer rate δ* Ref.[] H 15 1 Ref.[] (a) Displacement thickness (b) Shape factor Fig. 1 Distribution of the thickness of the boundary layer The Van Driest transformed mean velocities at location = 1 57 are shown in Fig. 13. gain, the results show good agreement with each other, satisfying both the linear law u + c = y+ and the log law u + c =.5 lny in the ranges y + 8 and y + 3, respectively. The mean profiles at locations = 1 and 113 in case, together with the results in Ref. [], are shown in Fig. 14. They almost fall on the same curve, satisfying turbulent flow similarity. Results in case are similar, so are not shown here. lthough in the above two cases, the results are correct no matter whether the Mach number modification is applied or not in the calculations of the density and temperature fluctuations for the inflow condition, when the real Mach number is appreciably larger than the Mach number of the flow field from which the inflow condition is deduced, the computation may fail. In that case, Mach number modification is necessary.

13 Inflow boundary condition for DNS of turbulent boundary layers 997 u + c Ref.[] u + = y + u + =.5lny y + Fig. 13 Van Driest transformed stream-wise mean velocity y/δ* 3 1 Ref.[]_1 Ref.[]_113 _1 _113 y/δ* 3 1 Ref.[]_1 Ref.[]_113 _1 _ u/u ~ ~ ~ ~ e T/T e (a) Stream-wise velocity (b) Temperature Fig. 14 Comparison of the mean profiles 3 Conclusions (1) Results of the three test cases show that the inflow conditions of spatial-mode DNS for turbulent boundary layers on a cone can be deduced from the same flow field data of a single instant, obtained by a temporal-mode DNS for the turbulent boundary layer on a flat plate, even if their Mach numbers and wall temperature conditions are different. () If the Mach numbers and the wall temperature conditions are the same for the cone and flat plate, then the deduction of the inflow condition is straightforward. (3) If the oncoming Mach number and wall temperature condition are different for the cone and flat plate, the distributions of the temperature and density at the inlet, deduced from the data on the flat plate, need to be modified both for their means and fluctuations. (4) y this inflow condition, the turbulent flow will eperience a certain adjustment to reach its fully developed state, and the length for the adjustment reaches only about 1R downstream of the inlet. From the computation cost and computational stability point of view, the proposed method is superior to the normal DNS method. cknowledgements The results of temporal-mode DNS for turbulent boundary layer on a flat plate is provided by Doctor Huang Zhangfeng, to whom the authors are grateful. References [1] Dong Ming, Luo Jisheng. Mechanism of transition in a hypersonic sharp cone boundary layer with zero angle of attack[j]. pplied Mathematics and Mechanics (English Edition), 7, 8(8): DOI: 1.17/s

14 998 DONG Ming and ZHOU heng [] Dong Ming. Direct numerical simulation of a hypersonic blunt cone turbulent boundary layer at Mach 6 with zero angle of attack[c]. In: The 6th cademic Conference on Computational Physics at Northwest Region, Hanzhong, Shanni Province, pr. 8, 9 97 (in Chinese). [3] Dong Ming, Luo Jisheng. The influence of the turbulent statistical characteristic by cone effect in a supersonic boundary layer[j]. Chinese Journal of Theoretical and pplied Mechanics, 8, 4(3): (in Chinese). [4] Pirozzoli S, Grasso F, Gatski T. Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M =.5[J]. Physics of Fluid, 4, 16(3): [5] Li Xinliang, Fu Deun, Ma Yanwen. DNS of compressible turbulent boundary layer over a blunt wedge[j]. Science in China, Ser G, 5, 48(): [6] Lund T S, Wu X H, Squires K D. Generation of turbulent inflow data for spatially-developing boundary layer simulations[j]. J Comp Phys, 1998, 14(): [7] Huang Zhangfeng, Zhou Heng. Inflow conditions for spatial direct numerical simulation of turbulent boundary layers[j]. Science in China, Ser G, 8, 38(3): (in Chinese). [8] Tang Hongtao. The mechanism of breakdown in laminar-turbulent transition and the characteristics of turbulence in an incompressible boundary layer on a flat plate [D]. Ph D dissertation. Tianjin: Tianjin University, 7 (in Chinese). [9] Li X, Fu D, Ma Y. Direct numerical simulation of a spatially evolving supersonic turbulent boundary layer at Ma = 6[J]. Chin Phys Lett, 6, 3(6): [1] Guarini S E, Moser R D, Shariff K et al. Direct numerical simulation of a supersonic turbulent boundary layer at Mach.5[J]. J Fluid Mech,, 414(1):1 33. [11] White F M. Viscous fluid flow[m]. New York: McGraw-Hill, [1] Gatski T, Erlebacher G. Numerical simulation of a spatially evolving supersonic turbulent boundary layer[r]. NS/TM , NS Tech Memo,. [13] Maeder T, dams N, Kleiser L. Direct simulation of turbulent supersonic boundary layers by an etended temporal approach[j]. J Fluid Mech, 1, 49(1):