[american institute of aeronautics and astronautics 49th aiaa aerospace sciences meeting including...

American Institute of Aeronautics and Astronautics 092407

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Numerical Analysis of Wall Properties in Crossing-Shock-Wave/Turbulent-Boundary-Layer Interactions

Yufeng Yao1, Andrea Salin2 and Sing H. Lo3

School of Aerospace and Aircraft Engineering, Kingston University, London SW15 3DW, UK

Three-dimensional crossing shock-wave and turbulent boundary layer interactions have been studied by using computational fluid dynamics (CFD) approach. The configuration is double sharp fins mounted on a flat plate with a deflection angle of 7o × 7o, 11o × 11o, 15o × 15o respectively and a fixed fin throat width of 32 mm and a fin length of 192 mm. The Mach number of incoming supersonic flow is 3.92 and the unit Reynolds number is 88 × 106. The grid convergence studies were performed for weak interaction 7o × 7o, and strong interaction 15o × 15o cases on three successive grids ranging from coarse to fine. It was found that the computed static pressure distributions are agreed well on medium and fine grids; and the wall heat transfer coefficients from the fine grid have shown better agreement with the experimental measurements. Simulation of intermediate interaction 11o × 11o case is thus only performed on the fine grid. For all three cases, the computed static pressure and wall heat transfer coefficient variations along the throat middle line (TML) and three cross-section locations have shown fairly good agreement with the experimental measurements and computational results from other researchers. The near-wall flow topologies have also shown good qualitatively agreement with the experimental visualization. For the 15o × 15o case, the computed heat transfer coefficient has exhibited significant decrease around the interaction region, which was not observed in the experiments. Further studies have been focused on this strong interaction case and the turbulent models influences on the static pressure and heat transfer coefficient predictions have been investigated. It was found that the turbulence model has little influence on static pressure distribution, but has significant impact on heat transfer coefficient with the RNG k- model producing violated results and the shear stress transport and eddy viscosity transport models producing results in better agreement with the test data. Due to strong shock-wave boundary layer interactions, flow separation exists and this has been revealed by velocity vectors and skin friction coefficients. It was believed that the decrease of wall heat transfer coefficient is related with flow separation that might alter the near-wall temperature profile. Several temperature profiles have thus been plotted and confirmed this hypothesis.

Nomenclature

A = Maximum inlet width B = Minimum throat width Cf = Skin friction coefficient Ch = Heat transfer coefficient Cp = Heat capacity of air at constant pressure C1 = Saddle point N1 = Node point Qw = Wall heat flux

1 Reader, School of Aerospace and Aircraft Engineering, Kingston University, London SW15 3DW, AIAA Senior Member. 2 Graduate Student, School of Aerospace and Aircraft Engineering, Kingston University, London SW15 3DW AIAA Student Member. 3 Senior Lecturer, School of Aerospace and Aircraft Engineering, Kingston University, London SW15 3DW.

49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-860

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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Taw = Adiabatic wall temperature Tw = Wall temperature M = Free-stream Mach number Ncell = Number of cell elements Nx, Ny, Nz = Number of grid points along three Cartesian coordinates x, y, z Po = Total pressure p = Static pressure R = Reattachment Re = Reynolds number S = Separation To = Total temperature U, V, W = Cartesian velocity components U = Free-stream velocity x, y, z = Cartesian coordinates

= Free-stream density = Fin deflection angle = Boundary layer thickness = Boundary layer momentum thickness = Shock wave pattern while interacting with a boundary layer x = Length of the cell size in the x-direction y1, z1 = Length of the first cell above the wall in the y and z directions, respectively y1

+, z1+ = Length of the first cell above the wall in wall unit in the y and z directions, respectively

yTML = Length of the cell size in the y-direction along the throat middle line

CFD = Computational fluid dynamics EVT = Eddy viscosity transport I, II, III = Cross-section locations RANS = Reynolds-averaged Navier-Stokes RNG k- = Re-normalisation group k-SST = Shear stress transport SWBLI = Shock-wave boundary layer interaction TML = Throat middle line

I. Introduction HE problem of three-dimensional (3-D) crossing shock-wave interacting with a spatially developing turbulent boundary layer contains rich fundamental flow physics, simply because it involves two important phenomena, i.e. the shock-wave

of high-speed flow and the boundary-layer of low-speed flow. With the resurgence of interest in high-speed flight in recent years, the problem of viscous-inviscid interactions in aerodynamic flows is once again receiving wider attentions, not only due to the inherent complexity and richness of the flow physics, but also due to engineering challenges posed by associated to large flow separation problems which increase drag and aerodynamic heating. The shock-wave boundary layer interaction phenomena occur over a wide range of high-speed aerodynamic flows, such as supersonic/hypersonic air intakes, propelling nozzles at off-design conditions and deflected controls at supersonic/transonic speeds.

Over the past fifty-years, a great deal of research interest has been directed towards both two-dimensional (2-D) and simplified three-dimensional (3-D) shock-wave boundary layer interaction (SWBLI) issues. More recently, some efforts have been focused on understanding more complicated full 3-D interactions that can be generally studied using relatively simple configurations so that to isolate the pertaining flow physics from the complexity associated with the entire aircraft structures. The problem of 3-D crossing shock-wave interacting with a spatially developing turbulent boundary layer is one of benchmark cases to be widely used for laboratory researches. For example, previous investigations have adopted simple geometry configurations, namely the single sharp fin and the double sharp fins mounted on a flat plate. A configuration which has received a great deal of research attention in the last decade is shown in Fig. 1a. This configuration simulates a generic high-speed sidewall-compression inlet (e.g. SCRAMJET inlet), within which a pair of crossing shock waves interacts

T


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with an incoming supersonic/hypersonic turbulent boundary layer. One of the primary goals of an efficient air-breathing high-speed inlet design is to effectively control 3-D shock/turbulence interactions to provide the engine with high and nearly uniform total pressure flow. However, flow separation and unsteadiness can result in significant non-uniformities and substantial loss of total pressure that can be detrimental to the operation of the engine. Also, the concentrated high heat fluxes in some region of the inlet walls can cause serious structural damages and possibly inlet “un-start” phenomena. These demands further in-depth studies, experimentally and computationally.

The wind tunnel experiments [1] and the Reynolds-averaged Navier-Stokes (RANS) based numerical simulations [2, 3] have been attempted for the double fin configurations. Recent review papers by Knight et al. [4] and Zheltovodov [6] provided detailed summary of the state-of-the-art in this field of research. It has been concluded that cases of moderate to strong crossing shock interactions, heat transfer and skin friction coefficient distributions as well as flow topology and structures, show remarkable disagreement with experimental measurements. It was also observed that, the primary flow features (e.g. primary separation and reattachment lines, and global pressure distributions) are not dependent on turbulence models, but secondary flow features (e.g. secondary singular lines, local pressure distributions, heat fluxes and skin friction coefficients) are. This might be primarily due to the inability of RANS turbulence models to predict low-turbulent flow in terms of the vortices developing immediately downstream of the initial shock interactions. Further experimental and numerical studies are thus required to validate these findings.

In the present study, we are going to revisit three symmetrical double fin configurations assessed by Thivet et al. [3, 5]. Both grid convergence and turbulent model assessment will be carried out and comparisons of surface static pressure and heat transfer fluxes will be made. In particular, the weak interaction case 7° × 7° and the strong interaction case 15° × 15° will be used as the validation benchmarks for comparing static pressure, skin friction coefficient and heat transfer distributions due to available experimental data. Resulting from the intermediate interaction case 11° × 11° are also compared qualitatively to the 7° × 7° and 15° × 15° results, because of no available heat transfer data are present for the 11° × 11° case, to the authors’ knowledge. Particular emphasis will also be placed on the strong interaction 15° × 15° case, in terms of flow topology and turbulence model influence on static pressure and skin friction coefficients. One of the main aims of the present research program is to improve the generally over-predicted computed heat transfer distributions by RANS turbulence modeling.

II. Physical Problem and Numerical Treatments

Figure 1a gives a sketch of double sharp fins mounted on a flat plate with incoming supersonic flow through a narrow passage. Due to the deflection (wedge) angle of the fin leading-edge ( ), the supersonic incoming flow will deflect and generate a crossing shock wave at strength dependent on the free-stream Mach number and the fin deflection angle. For a sufficient large fin angle, the strength of the oblique shock wave will be stronger and the interactions with the undisturbed incoming turbulent boundary layer will often lead to a flow separation region along the shock line on the plate (often denoted as the -shock system). At the rear of the fin near the trailing-edge, the flow will undergo the Prandtl-Meyer expansion process, similar to that seen in the expansion-compression corner case. The 3-D double sharp fins case will have additional complicity with the oblique shock-shock interactions throughout the flow passage at downstream location and for symmetrical double-fins; this will happen along the throat middle line (TML). At meantime, the 3-D shock-wave and turbulent boundary layer interactions exist along the shock-wave lines. This leads to complex flow structures in the near wall region and consequently it will have significant effect on key flow parameters such as skin friction and heat transfer characteristics.

A. Configurations

A total of three symmetrical double fin configurations with deflection angles ( 1 = 2) of 7°, 11° and 15° are numerically studied in the present work. Ranging from weak to strong crossing shock-wave interactions, they are namely the 7° × 7°, 11° × 11° and 15° × 15° double fin cases thereafter. Following the published work by Thivet et al. [3, 5], a flow passage domain between the two fins is set up as seen in Figure 1b, where some key geometrical parameters are provided. In the corresponding experiments [1], two symmetrical chamfered sharp fins of 192 mm in length and 100 mm in height are mounted in a flat plate. The minimum throat distance (B) is kept as a constant of 32 mm and it is independent of the wedge


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angle change. Hence the maximum inlet width (A) for the 7° × 7°, 11° × 11° and 15° × 15° cases are 71.4 mm, 75.6 mm and 79.2 mm, respectively. Same as Thivet et al. [3, 5], a shorter height of 80 mm is used in the current study, since no significant vertical gradients were observed between y = 70 – 90 mm. Other dimensions described in above remain exactly the same as the reference papers [1-3]. Figure 1b also illustrates the throat middle line (TML) along the bottom wall surface and three cross-section locations as x = 46 mm, 79 mm and 112 mm measured from the leading edge of the fins (marked as I, II and III in Fig. 1b, respectively), where results comparisons with available experimental measurements will be made.

Figure 1. (a) Sketch of flow over double sharp fins model; (b) three fin passage domains for present simulations (all dimensions are in millimetre).

B. Flow Conditions, Computational Domain and Numerical Model

The flow conditions are the incoming Mach number (M ) of 3.92, the total pressure (Po) of 1485 kPa, and the total temperature (To) of 260 K, yielding a unit Reynolds number (Re) of 88 × 106. These parameters are taken from the experiments [1]. During the tests, the leading edge of the fins were placed at 14 mm downstream of a location, where undisturbed turbulent boundary layer develops with boundary layer thickness ( = 3.5 mm) and momentum thickness ( = 0.128 mm) measured in absence of the fins geometry. To ensure that the numerical predicted undisturbed turbulent boundary layer has similar properties at the same location, a precursor simulation of a spatially developing turbulent boundary layer along a flat plate was carried out and based on the evaluation of boundary layer properties; a distance of 168 mm between the leading edge of the flat plate (i.e. the inlet of the computational domain) and the leading edge of the fins (i.e. the entrance of passage flow) was chosen. This will result in a computational domain length in the x-direction of 360 mm. Due to the geometry symmetry of the problem, and also considering the strength of shock-wave and boundary layer would not reach a level of unsteady flow in the lateral spanwise direction, it was then decided to consider only half a domain in the spanwise direction, in order to save the computational cost. The final computational domain has a dimension of 360 mm × 80 mm in the streamwise (x) and the wall-normal (y) directions with a varied spanwise length of 71.4 mm – 79.2 mm, depending on the deflection angle of the fins. The origin of the coordinate system ‘O’ is located at the middle of the passage entrance plane, i.e. the cross point of the TML and the streamwise location of the fins leading edge.

Numerical models are considering the Reynolds-averaged Navier-Stokes (RANS) based computational fluid dynamics (CFD) approach, in which three-dimensional compressible flow governing equations are solved together with a turbulence model as a closure. For three configurations considered, both the flow and shock-wave unsteadiness are not significant. Hence, all simulations are run in steady manner at this stage with unsteady simulations planned in further stages. As for the


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highest deflection angle of 15o, previous study of Thivet et al. [3, 5] revealed a flow separation region at the TML. Thus, Menter’s shear stress transport (SST) turbulence model which is better suitable for separated flow is adopted. This model will be used for all simulations, unless otherwise stated. Further assessment on turbulence model influence is carried out for the 15° × 15° case, using the one-equation eddy viscosity transport (EVT) model and the two-equation RNG k- model. The boundary conditions are a uniform supersonic inflow of M = 3.92 at the inlet plane, and supersonic outflow conditions at the outlet plane. A symmetric condition is used at the side planes upstream of the fin leading edge. The top surface uses free-slip condition and the bottom surface and the fin walls use no-slip condition. The simulations initially use an adiabatic wall condition for the bottom surface and fin walls and compare the flow field properties (e.g. static pressure) with available test data. Later, an isothermal wall condition with a fixed temperature of 270 K is used in the simulations that allow the evaluation of heat transfer coefficient on the wall surfaces. In general, steady flow simulations will run until a convergence criterion of about 10-5 in terms of root-mean-square (RMS) residual achieved. The default total number of iterations is set to be 250, and normally the longest simulation of the 15° × 15° case on a fine grid requires total computational time of about 90 hours on a desktop P4 computer.

III. Meshing and Grid Refinements

By referencing to the previous study of Thivet et al. [3], three structured meshes were generated, namely a coarse grid, a medium grid and a fine grid. In each case, the grid size is kept uniform in the streamwise direction with a constant cell length ( x) of 2 mm used, and the grid is highly stretched in the wall normal (y) and the spanwise (z) directions, respectively. The coarse grid has 180 × 20 × 20 points in the streamwise, the wall normal and the spanwise directions. The number of grid points in the streamwise direction Nx is kept the same, while further refining twice the grid points in both the wall normal, and the spanwise directions, respectively. This will result in three structured meshes for each fin configurations, leading to a total of nine grids. It needs to point out that in order to obtain accurate results particularly the heat transfer coefficient, the near-wall mesh resolution has to be fine enough to capture the accurate temperature profile inside the thermal layer.

Table 1 gives a summary of the coarse, medium and fine grids in terms of number of cells (Ncell), number of grid points (Nx, Ny, Nz) in the streamwise (x), the wall normal (y) and the spanwise (z) directions, respectively. The first cell heights from the bottom wall ( y1

+), the fin side-wall ( z1), and at the symmetry (x-y) plane ( zTML) are also presented. For example, the fine grid of the 15° × 15° case has about 48 grid points located inside the boundary layer thickness at a location of x = 168 mm, yielding the height of the first cell to the bottom wall surface about 0.1 in wall unit. In order to capture shock-shock interactions around TML, finer grids are also generated nearby. In comparing to previous studies by Thivet et al. [3], same near-wall resolutions are considered in present simulations.

Case Grid Ncell Nx / x (mm) Nz × Ny y1 ( m) y1+

(TML) z1 ( m) zTML (mm)

7° × 7°

Coarse

Medium

Fine

72,000

288,000

1,152,000

180/2

180/2

180/2

20 × 20

40 × 40

80 × 80

4.9

2.1

1.1

1.2 - 4.7

0.5 - 2.1

0.2 - 1.1

12.1

5.3

2.8

2.70

1.32

0.65

11° × 11°

Coarse

Medium

Fine

72,000

288,000

1,152,000

180/2

180/2

180/2

20 × 20

40 × 40

80 × 80

4.9

2.1

1.1

1.2 - 7.7

0.5 - 3.5

0.2 - 1.8

12.6

5.7

2.8

2.65

1.26

0.64

15° × 15°

Coarse

Medium

Fine

117,000

468,000

1,872,000

180/2

180/2

180/2

25 × 26

50 × 52

100 × 104

4.5

2.1

0.6

1.8 - 9.2

0.8 - 3.9

0.1 - 1.1

11.3

5.6

2.7

0.69

0.41

0.19

Table 1. Simulation cases and grid parameters.


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IV. Results and Discussions

Numerical simulations start with grid refinement study for the weak interaction 7o × 7o case and the strong interaction 15o

× 15o case, and results are compared with experimental measurements [1] and computations [3]. Based on these studies, the intermediate interaction 11o × 11o case was carried out only on the fine grid. Results from all three cases are compared to available experiments and computations. As previous explained, simulation uses an adiabatic wall condition at first and the comparisons of static pressure distributions along the TML and at three cross-sectional locations of I, II, III are made. In order to evaluate the wall heat transfer coefficient, simulations use an isothermal wall condition is considered. The effect of turbulence model on the heat transfer characteristics has been studied for the 15o × 15o case by comparing three turbulence models. Finally, flow separation, skin friction coefficient distributions and some flow topologies are presented.

A. Pressure and heat transfer coefficient distributions

Figures 2 and 3 have shown the static pressure and the heat transfer coefficient distributions, respectively for the 7° × 7°, 11° × 11°, and 15° × 15° configurations. In both figures, the first and the second columns are the grid refinement studies of the 7° × 7° and the 15° × 15° cases and the third column presents the fine grid results for all three 7° × 7°, 11° × 11°, and 15° × 15° cases. The heat transfer coefficient (Ch) is evaluated using a formula as:-

Ch = Qw/( U Cp(Tw-Taw)) , (1)

where , U are free-stream quantities, Cp is heat capacity of air at constant pressure, Taw is computed adiabatic wall temperature from adiabatic simulations, while Qw is computed wall heat flux corresponding to the constant wall temperature Tw of 270 K from isothermal simulations.

B. The 7° × 7° Configuration

Figure 2 depicts the computed static pressure distributions from three grids of coarse, medium and fine in comparison with measurements [1]. The inviscid pressure distributions from isentropic shock relations and the computational results of Thivet et al. [3] are also presented for reference. By comparing these results, it was found that good mesh convergence has been achieved between the medium and the fine grids, due to a near clasp of static pressure distributions. Overall, the computed surface static pressure distributions are in good agreement with the experimental data along the TML and three cross section locations I, II, III. Figure 2(a3) also shows that the flow is compressed after the primary shock-shock interaction at location II, in correspondence of the TML and two reattachment lines R1, R2 closed to the fin walls. Further downstream, pressure continues to increase along the TML reaching a peak value at location III (Figure 2(a4)). One important experiment observation [1] previously was the recovery of static pressure at the middle of the flow passage around x = 79 mm, which has been successfully captured by present simulations (Fig. 2(a3)). Overall, present results on the fine grid are in good agreement with the experimental measurements, similar to that of Thivet et al. [3].

Figure 3 compares the computed and measured heat transfer coefficient distributions from three grids from coarse to fine. Fig. 3(a1) shows that along the TML, the computed heat transfer coefficient Ch is in good agreement with the measurement until about x = 80 mm; after that, present simulation slightly under-predicts the Ch. This is quite similar to that predicted by Thivet et al. [3], in which under-prediction of Ch is shown in downstream region, but less significant after x = 135 mm. The grid convergence of Ch is also assessed (not shown here). At the three cross sections I, II, III (Figs. 3(a2)-3(a4)), the computed Ch exhibits high peak values in correspondence of the reattachment lines near the fin side walls. It has been seen that while reasonable good agreement in the spanwise direction has been achieved, some under-predictions exist in the region close to the TML, while over-predictions away from it by a factor of two. Overall present results have shown very good agreement with the experimental data [1] and the computational results of Thivet et al. [3].


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a1) b1) c1)

X [mm]

P/P

1

0 50 100 150 2000

1

2

3

4

5

6

ExperimentInviscidCoarseMediumFineThivet et al.

7°x 7°

TML

X [mm]P

/P1

0 50 100 150 2000

2

4

6

8

10

12

ExperimentInviscidCoarseMediumFineThivet et al.

15°x15°

TML

X [mm]

P/P

1

0 50 100 150 2000

2

4

6

8

10

12

Experiment 7°x7°Fine 7°x7°Experiment 11°x11°Fine 11°x11°Experiment 15°x15°Fine 15°x15°

TML

a2) b2) c2)

Z [mm]

P/P

1

-30 -20 -10 0 10 20 30

1

2

3

4 x=46mm

7°x 7°

R1 S1 S2 R2

Z [mm]

P/P

1

-30 -20 -10 0 10 20 30

2

4

6

8

10 x=46mm

15°x15°

R1 S3 S4 R2

Z [mm]

P/P

1

-30 -20 -10 0 10 20 30

2

4

6

8

10 x=46mm

a3) b3) c3)

Z [mm]

P/P

1

-30 -20 -10 0 10 20 30

1

2

3

4 x=79mm

7°x 7°

R1 S1 S2 R2

Z [mm]

P/P

1

-30 -20 -10 0 10 20 30

2

4

6

8

10 x=79mm

15°x15°

R1 S5 S6 R2

Z [mm]

P/P

1

-30 -20 -10 0 10 20 30

2

4

6

8

10 x=79mm

a4) b4) c4)

Z [mm]

P/P

1

-30 -20 -10 0 10 20 30

1

2

3

4 x=112mm

7°x 7°

R1 S3 S4 R2

Z [mm]

P/P

1

-30 -20 -10 0 10 20 30

2

4

6

8

10 x=112mm

15°x15°

R1 S5 S6 R2

Z [mm]

P/P

1

-30 -20 -10 0 10 20 30

2

4

6

8

10 x=112mm

Figure 2. Static pressure distributions along the TML and at three cross-section locations I, II, III.


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a1) b1) c1)

X [mm]

Ch

x10

3

0 50 100 150 2000

1

2

3

4

5

6ExperimentFineThivet et al.

7°x 7°

TML

X [mm]

Ch

x10

3

0 50 100 150 2000

1

2

3

4

5

6

ExperimentCoarseMediumFineThivet et al.

15°x15°

TML

X [mm]

Ch

x10

3

0 50 100 150 2000

1

2

3

4

5

6

Experiment 7°x7°Fine 7°x7°Fine 11°x11°Experiment 15°x15°Fine 15°x15°TML

a2) b2) c2)

Z [mm]

Ch

x10

3

-30 -20 -10 0 10 20 300

1

2

3

4

5

6 x=46mm

7°x 7°

R1 S1 S2 R2

Z [mm]

Ch

x10

3

-30 -20 -10 0 10 20 300

1

2

3

4

5

6 x=46mm

15°x15°

R1 S3 S4 R2

Z [mm]

Ch

x10

3

-30 -20 -10 0 10 20 300

1

2

3

4

5

6 x=46mm

a3) b3) c3)

Z [mm]

Ch

x10

3

-30 -20 -10 0 10 20 300

1

2

3

4

5

6 x=79mm

7°x 7°

R1 S1 R2S2

Z [mm]

Ch

x10

3

-30 -20 -10 0 10 20 300

1

2

3

4

5

6 x=79mm

15°x15°

R1 S5 S6 R2

Z [mm]

Ch

x10

3

-30 -20 -10 0 10 20 300

1

2

3

4

5

6 x=79mm

a4) b4) c4)

Z [mm]

Ch

x10

3

-30 -20 -10 0 10 20 300

1

2

3

4

5

6 x=112mm

7°x 7°

R1 S3 R2S4

Z [mm]

Ch

x10

3

-30 -20 -10 0 10 20 300

1

2

3

4

5

6 x=112mm

15°x15°

R1 S5 S6 R2

Z [mm]

Ch

x10

3

-30 -20 -10 0 10 20 300

1

2

3

4

5

6 x=112mm

Figure 3. Heat transfer coefficient distributions along the TML and at three cross-section locations I, II, III.


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C. The 15° × 15° Configuration

Figure 2(b1-b4) shows a comparison of the computed and the measured static pressure distributions for the 15° × 15° case. They are also compared with Thivet et al. fine grid results [3] and the calculated inviscid pressure distributions. Same as that for the 7° × 7° case, the grid refinement study is performed, showing that mesh convergence occurred between the medium and the fine grids due to the clasp of static pressure distributions. The computed static pressure distributions are also in good agreement with the measurements along the TML until about x = 70 mm with the pressure increase where the first crossing shock wave interaction. After that location, simulation over-predicts and this agrees with a previous study by Thivet et al. [3] (Figure 2(b1)). Both simulations predict the peak pressure with a correct streamwise location at x = 98 mm, but at a magnitude higher than the measurements [1]. Downstream at x = 132 mm, present computations under-predict the static pressure recovery. It is probably due to the under-prediction of the expansion fan resulting from the side wall angles (Figure 2(b1)). Further visualization as shown later in Figure 6(a3) indicates that the first cross section (I) is located exactly on the fluidic throat observed in the experiment. However, this fluidic throat has not so far been captured in computations (Figure 6(b3)) in which a saddle point C1 is located between x = 48 – 49 mm compared to x = 50 – 51 mm in the experiment. A node point N1 is also predicted upstream at about x= 67 – 68 mm compared to x = 68 – 69 mm of the experiment. Because of these differences, the computed results tend to over-predict the static pressure around the TML between the separation lines S5 and S6. There are two weak secondary peaks can be seen in a region near the TML and they are corresponding to two reattachment lines R1 and R2. At cross sections I and II in the spanwise direction, Fig. 2(b2-b3) shows that present static results exhibit the agreement with the measurements slightly better than those of Thivet et al. [3]. However, poorer agreement is noticed at the cross section III, where present computations significantly over-predict the static pressure between the separation lines S5 and S6 near the TML.

The separation region between two singular (saddle and node) points C1 and N1 observed in both the experiment (Figure 6(a3)) and the present computation (Figure 6(b3)), is also evident in the Ch distributions. Compared to the measured value, influence on the computed Ch can still be observed in Fig. 3(b1) at x = 48 – 67 mm. However, the trend is remarkably different from the experiment, as the Ch distributions under-predict the measurement data in this region. There are three peaks shown in the present computed Ch distributions (Figure 3(b1)) and they are actually corresponding to the locations of pressure peaks at the node point N1 location (x = x= 67 – 68 mm), at x = 109 mm and further downstream at x = 146 mm, respectively. The predicted maximum Ch magnitude is about twice of the measured value, and it is located downstream at x = 98 mm, where the primary pressure peak occurs. Comparing to the test data, present computations predict the heat transfer coefficient distribution slightly better than that of Thivet et al. [3] between x = 84 - 103 mm, but over-predict the maximum peak value. The possible reasons of under-predicting the Ch distribution at the crossing shock interaction region will be further discussed in later. Figure 3(b2) shows that computed results at the first cross section I significantly under-predict both the measurement and the computation of Thivet et al. [3] near the TML region and in proximity of the two separation lines S3 and S4 (Figure 6(b3)). The Ch distributions at other two cross sections (II and III) in Figure 3(b3-b4) also present significant difference between the measurements and the computational results of Thivet et al. [3]. In fact, computations predict a higher peak near the TML between the two separation lines S5 and S6 that was not observed in the experiment. According to Thivet et al. [3], this may be due to either the well-known numerical carbuncle-phenomena or poor measurement accuracy, or most probably a combination of both. In addition, a weak second Ch peak intensifying towards the fin wall is predicted (Fig. 3(b4)). In general, the present computed results are in fairly good qualitative and quantitative agreement with the Thivet et al. predictions [3]. Nevertheless, the computed Ch distributions are generally found to over-estimate the experiment by a factor of two. According to Thivet et al. [3, 5], the reason of this over-prediction is probably related to the hypothesis that two-equation turbulence models may produce excessive turbulence kinetic energy (TKE) level in the outer part of the boundary layer as it crosses the shock waves. These high TKE levels can be convected to the walls by the vortices downstream of the shock and stimulate the heat transfer increase. Although different realizability corrections to decrease TKE levels have been tested by Thivet et al. [5], there are still little improvements on reducing the heat transfer production levels. Hence the difference between the computations and experiment is still retained so far. To improve the prediction may require high level of numerical approach such as large-eddy simulation.

D. The 11° × 11° Configuration

Figure 2(c1-c4) and Figure 3(c1-c4) show a comparison of static pressure and heat transfer coefficient (Ch) distributions from fine grid simulations of 7° × 7°, 11° × 11° and 15° × 15° cases, respectively. For the intermediate interaction 11° × 11° case, the shock boundary layer interaction level is not very strong and the RANS predicted static pressure distributions are


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overall in good agreement with the experimental data along the TML (Figure 2(c1)) and at the three cross sections I, II, III (Figure 2(c2-c4), respectively. However, due to the lack of experimental measurement data of heat transfer coefficient, the Chdistributions are merely referenced to other two cases of 7° × 7° and 15° × 15°. Similar to that of 15° × 15° case, results show significant increase of Ch after the interaction region and the spanwise variations are also very similar to that of 15° × 15° case (Figure 3(c1-c4)). Same as two other cases, mesh refinement for 11° × 11° is shown in Table 1 and simulation has only been performed on the fine grid results for reasons previously explained.

E. Effects of Turbulence Model

The effect of turbulence model on the prediction is assessed for the 15° × 15° fine grid case and the discussion is given in this section. The static pressure and heat transfer coefficient (Ch) distributions are shown in Figure 4(a1-a4) and Figure 4(b1-b4), respectively. In addition to the shear stress transport (SST) turbulence model, two additional turbulence models are adopted and they are the one-equation eddy viscosity transport (EVT) model, and the two-equation RNG k- model.

As shown in Figure 4(a2-a4), the EVT model does improve the predicted static pressure distributions at two cross sections I and III. In particular, better predictions of static pressure distributions are shown at the cross section I between the separation lines S3 and S4 (Figure 4(a2)). It also captures the pressure level very well at the reattachment lines R1 and R2 at the cross section III (Figure 4(a4)). Most importantly, the EVT model is found to successfully predict the Ch distributions along the TML. It predicts the maximum Ch peak at x = 98 mm about 20% lower compared to that by the SST model, that means more close to test data. The Ch distributions at the three cross sections present some additional features. While computations still predict a high peak near the TML between two separation lines S5 and S6 same as that by the SST model, five secondary peaks are now captured at the cross section II (Figure 4(b3)). However, at the cross section III, the Ch value has been significantly over-estimated between the reattachment liens R1 and R2.

The RNG k- model slightly over-predicts the x-location of pressure rises due to the shock interaction (Figure 4(a1)). Although the primary peak static pressure along the TML has been significantly over-predicted in comparison to other two models, the pressure recovery due to the expansion fan at about x = 130 mm is correctly predicted. Figure 4(b1) compares the Ch distributions of all three turbulence models along the TML. The RNG k- is the only model that agrees with the experiment Ch results until about x = 62 mm (despite the accuracy of the test data might be arguable, see discussion in later). After this x-location, the Ch distributions exhibit very strange behaviour with three high peaks as shown in Figure 4(b1), indicting certain model instability. Interestingly, downstream after x = 115 mm, the Ch recovers and agrees well with the test data. At the three cross sections, the Ch distributions are generally flattened at a low magnitude by at most a factor of two compared to the EVT and the SST predictions. The computations still predict a high peak near the TML between two separation lines S5 and S6 at cross sections II and III (Figure 4(b3-b4)).

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Figure 4. Turbulence model assessment study for the 15° × 15° fine grid case (a1-a4) static pressure distributions, and (b1-b4) heat transfer coefficient distributions, along the TML and at three cross sections I, II, III.

F. Skin friction coefficients

Figure 5a shows the skin friction coefficient (Cf) distributions along the TML for the 7° × 7°, 11° × 11° and 15° × 15° fine grids simulations. Velocity vectors in the x-y symmetric plane from the 15° × 15° fine grid simulation is superimposed at the background to show the flow separation region due to strong shock boundary layer interactions. It is evident that the Chdecrease as noted previously in Figure 3(b1) at x = 48 – 67 mm (i.e. the x-location of saddle point C1 and node point N1,


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respectively) is closely associated with the flow separation region where skin friction coefficients have negative values. There is no flow separation observed for the 7° × 7° case and for 11° × 11° case, the flow separation is merely incipient.

Figure 5b compares the skin friction coefficient distributions of the 15° × 15° case from three different turbulence models as mentioned in previous section. The experimental data [6] are used for comparison. It can be seen that turbulence model has significant effect on the skin friction distribution both inside the separation region and downstream after the reattachment. Both the SST and the EVT models have captured the flow separation, despite that the position of saddle point C1 and node point N1 in the Cf distribution curve is shifted of about 9 mm downstream in the EVT model compared to that of SST model. This leads to some differences in the flow topology around the interaction region. The RNG k- did not predict the flow separation at all. Among three turbulence models, the EVT model seems to be the most promising one in terms of the skin friction distributions, as its predictions are in better agreement with the experimental measurement compared to other two models. As previously discussed, the heat transfer variation around the interaction region might also be closely related on the flow separation in terms of size and shape, thus it is crucial for accurately predicting the flow separation in the shock boundary layer interaction process.

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Figure 5. (a) Skin friction coefficient along the TML from the 7° × 7°, 11° × 11° and 15° × 15° fine grids simulations, superimposed with velocity vectors on a symmetric x-y plane from the 15° × 15° fine grid simulation. (b) skin friction coefficient from different turbulence model in comparison with experimental measurement data [6].

G. Flow Topologies

The flow at the channel entrance of the fin leading edges is determined by the interactions of the swept shock waves generated by the sharp fins and a supersonic turbulent boundary layer on the flat plate. Due to the strong inviscid primary shocks, skin friction lines initially converge to the limiting streamlines of two primary separation lines S1 and S2 and reattach to the reattachment lines R1 and R2, forming a pair of counter-rotating vortices. This incoming bifurcated shock structures merge on the centreline in a large vertical region, becoming distorted but maintaining a typical -shaped shock structure. Some streamlines penetrating downstream along the TML converge to form a characteristic throat called ‘fluidic’ throat.

Figure 6 shows comparisons of the experimental oil-film visualization (Fig. 6(a1-a3)) and the computed skin friction streamlines (Fig. 6(b1-b3)) at the bottom wall surface for the 7° × 7°, 11° × 11° and 15° × 15° fine grid configurations. Results have shown that the computed skin friction streamlines at the bottom wall for the 7° × 7° and 11° × 11° cases are in qualitatively good agreement with the experimental oil-flow visualisations as seen in Figure 6(a1-b1) and Figure 6(a2-b2), respectively. For the 7° × 7° case, the computed ‘fluidic’ throat is also observed likewise in the experiment and no flow separation is visible. For the 11° × 11° configuration, due to the moderate shock interaction, a small flow separation region


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between C1 and N1 was observed in the experiment. However, the simulation has only captured merely an incipient flow separation bounded by the two separation lines S5 and S6 and both the saddle point C1 and the node point N1 are not visible (Figure 6(b2)). For the 15° × 15° case, Figure 6(b3) shows that the primary separation lines S1, S2 intersect the centreline node point N1, and no fluidic throat is observed in the computation, in contrary to the experiment (Figure 6 (a3)). A large scale flow separation region bounded by two separation lines S5 and S6 is more detained at the throat behind the node point N1. Results exhibit a more compressed and smaller flow separation laterally in the central region. The flow separation is clearly observed in the interaction region between the saddle point C1 and the node point N1, consistent with that shown in Fig. 5a and Fig. 8a in later. The separation and reattachment lines are clearly visible and are in qualitative good agreement with the experiment. However, the separation line S7 and S8 are not captured by the simulations, probably due to poor local grid resolution.

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Figure 6. Comparisons of experimental visualization and the computed streamlines. (a1-a3) oil-film visualization of experiment [1] and (b1-b3) computed skin friction streamlines on the bottom wall surface for the 7° × 7°, 11° × 11° and 15° × 15° configurations, respectively. The fine grids results are presented with the SST turbulence model.

Figure 7(a-c) represents the computed skin friction and heat flux on the bottom wall surface of the 7° × 7°, 11° × 11° and 15° × 15°, respectively. The upper-half is the computed heat flux contour, while the lower-half is the skin friction coefficient contour. It can be seen that as the strength of shock-wave boundary layer interaction increases, the area of low value (or even negative value in case of flow separation) of skin friction coefficients tend to cluster along the crossing-shock interaction region, where adverse pressure gradients exist and when it is strong enough, it will cause boundary layer flow to separate. As shown in Figure 7 and Figure 8, the heat transfer fluxes also decrease in these areas, in particular inside the flow separation region. The maximum skin friction and heat transfer on the bottom wall are found to be in downstream of the interaction


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region, corresponding to two reattachment lines R1 and R2. Figure 8 shows the 15° × 15° simulation results with the region characterized by strong adverse pressure gradients where low levels of heat fluxes as well as negative skin friction coefficients are evident.

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Figure 7. Comparisons of computed skin friction and heat flux on the bottom wall of (a) 7° × 7°, (b) 11° × 11° and (c) 15° × 15°, respectively. For each graph, the upper-half presents computation results of heat flux and the lower-half presents computation results of skin friction coefficient. Note that the fine grid simulation results with the SST turbulence model are presented.

Figure 8. Comparisons of computed skin friction and heat flux on the bottom wall. Upper-half: computed low heat flux value of 0 – 5000 W m-2. Lower-half: computed negative skin friction coefficient (i.e. flow separation region). Note that the fine grid simulation results with the SST turbulence model are presented.

In order to investigate the lower heat flux inside the flow separation region, Fig. 9 shows flow details around the crossing-shock interaction region. On the bottom wall surface, the negative skin friction region is shown with superimposed streamlines. In the symmetry x-y plan, temperature contours are presented, showing the thermal boundary layer in correspondence of the shock interaction with superimposed velocity vectors. The flow structure has indeed shown its complexity near the interaction region. As the lower heat fluxes are related to temperature profiles particularly the near wall slope. Figure 10 gives the temperature profiles of seven cross-section locations along the TML. These locations are selected based on the peaks and key locations of the Ch distribution, in order to understand the causes of the heat transfer coefficient decrease between the saddle point C1 and the node point N1. The profiles are indeed shown significant changes in the near wall shape and its slope. Further quantification is necessary and it will be the subject of future study.

Figure 9. Flow details around the recirculation region in the symmetric (x-y) plane where temperature contours are shown with superimposed velocity vector) and the bottom wall surface (y=0) where negative skin friction contours are shown with superimposed streamlines. Note that the fine grid simulation results with the SST turbulence model are presented.


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Figure 10. Computed static temperature profiles at seven x-locations from the 15° × 15° fine grid case (with the SST turbulence model), showing the variations of near wall slope. The horizontal coordinate is shifted by 300 K for each successive graph.

V. Conclusions

Numerical studies of three-dimensional crossing shock-wave and turbulence boundary-layer interactions have been performed by using the computational fluid dynamics (CFD) approach based on solving the Reynolds-averaged Navier-Stokes (RANS) equations with turbulence model as a closure. The configuration considers double sharp fins mounted on a flat plate with a deflection angle of 7o × 7o, 11o × 11o, 15o × 15o respectively and a fixed fin throat width of 32 mm and a fixed fin axial length of 192 mm. The Mach number of incoming supersonic flow is 3.92 and the unit Reynolds number is 88 × 106. The grid convergence studies are carried out on three successive grids ranging from coarse to fine. For all three cases, the computed static pressure along the throat middle line (TML) and the traverse line at three streamwise locations are in fairly good agreement with the experimental measurements and other computational results. The near-wall flow topologies have shown good qualitatively agreement with the experimental visualization. However for the 15o × 15o case, the computed heat transfer coefficient (Ch) has exhibited significant decrease around the interaction region, which does not agree with the experimental measurements. Using different turbulence models have shown little changes on static pressure, but the models do have considerable effects on heat transfer coefficient. The computed heat transfer coefficient however still under-predicts in the interaction region and over-predicts in downstream. The under-prediction of the Ch distribution is found to be related with the flow separation that might influence local temperature profile with low near wall heat fluxes, while the over-prediction of the Ch in downstream is most likely related to the limitation of current turbulence models used in the RANS approach. Computation using more advanced large-eddy simulation method would be desirable, especially for strong shock-wave boundary layer interaction scenarios.

Acknowledgments

The second author would like to acknowledge the Graduate Teaching Assistantship provided by Kingston University.

References

1 Zheltovodov, A.A., Maksimov, A.I., and Shevchenko, A.M., “Topology of Three-dimensional Separation under the Conditions of Symmetric Interaction of Crossing Shocks and Expansion Waves with Turbulent Boundary Layer,” Therm. Aero Mech., 5(3), 293-312, 1998.

2 Zheltovodov, A.A., Maksimov, A.I., Schülein, E., Gaitonde, D., and Schmisseur, J.D., “Verification of crossing-shock-wave/boundary layer interaction computations with the k– turbulence model,” Proceedings of the International Conference on the Methods of Aero-physical Research (ICMAR’2000), Part I, Novosibirsk, Russia, pp. 231–241, 2000.


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3 Thivet, F., Knight, D.D., Zheltovodov, A.A. and Maksimov, A.I., “Analysis of Observed and Computed Crossing-Shock-Wave/Turbulent-Boundary-Layer Interactions,” Aerospace Science and Technology, 6(1), 3-17, 2002.

4 Knight, D.D., Yan, H., Panaras, A.G. and Zheltovodov, A.A., “Advances in CFD Prediction of Shock Wave Turbulent Boundary Layer Interactions,” Progress in Aerospace Sciences, 39(1-2), 121-184, 2003.

5 Thivet, F., Knight, D.D., Zheltovodov, A.A. & Maksimov, A.I., “Insights in turbulence modelling for crossing-shock-wave/boundary-layer interactions,” AIAA Journal, 39(6), 985-995, 2001.

6 Zheltovodov, A.A., “Some advances in research of shock wave turbulent boundary layer interactions,” AIAA paper 2006-496, 2006. 7 Garrison, T.J., Settlest, G.S., Narayanswami, N. & Knight, D.D., “Laser Interferometer Skin-Friction Measurements of Crossing-

Shock-Wave/Turbulent-Boundary-Layer Interactions,” AIAA Journal, 32(6), 1234, 1994. 8 Garrison, T.J, “The interaction between crossing-shock waves and a turbulent boundary layer,” PhD Thesis, Penn State University,

1994. 9 Gaitonde, D. & Shang, J.S., “Structure of a turbulent double-fin interaction at Mach 4,” AIAA Journal, 33(12), 2250-2258. 1995.

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