This study is devoted to investigating a flow around a stationary or moving sphere by using direct numerical simulation with immersed boundary method (IBM) for the threedimensional compressible NavierStokes equations. A hybrid scheme developed to solve both shocks and turbulent flows is employed to solve the flow around a sphere in the equally spaced Cartesian mesh. Drag coefficients of the spheres are compared with reliable values obtained from highly accurate boundaryfitted coordinate (BFC) flow solver to clarify the applicability of the present method. As a result, good agreement was obtained between the present results and those from the BFC flow solver. Moreover, the effectiveness of the hybrid scheme was demonstrated to capture the wake structure of a sphere. Both advantages and disadvantages of the simple IBM were investigated in detail.
Acoustic wave from rocket nozzle on the rocket launch may critically damage satellites inside the payloads of the rocket since the wave sometimes becomes quite strong. The prediction of the crucial acoustic field around rocket launching sites has been conducted by the past enormous studies and testing data [
Fukuda et al. showed that acoustic waves are primarily attenuated by the interactions between particles and turbulence [
We develop the Cartesian flow solver with an immersed boundary method (IBM) [
The governing equations of the present flow simulations are threedimensional compressible NavierStokes equations:
All variables are normalized by the freestream values of the density, the sound speed, and the reference length. The above equations are discretized on an equally spaced Cartesian mesh with cellcentered arrangement. To eliminate additional numerical dissipation everywhere, except in the vicinities of shock waves and potential flows, the inviscid terms are computed by a hybrid scheme that combines the thirdorder monotone upstreamcentered scheme for conservation laws (MUSCL) Roe scheme [
In our previous studies, we proposed the hybrid scheme estimated by the Ducros sensor [
Here,
Here a value
In the previous study [
A simple IBM based on the level set function is used to represent object boundaries in this study. The level set function is determined as a signed distance of minimal distance from the object surface. In the case of multiple objects, multiple level set functions are calculated based on simple minimum distance approach [
Figure
Schematic of the present IBM for a sphere.
Aerodynamic forces acting on the object surface as pressure and friction can be evaluated on the cell face between the fluid and the ghost cells. In the present method, a simple algorithm where the surface polygons are not necessary to estimate the forces by using staircase representation is employed. The blue and white cells correspond to the fluid and ghost cells in Figure
Representation of sphere for force evaluation.
The reflection of sound waves from outer boundaries may affect the flow in the subsonic case. Therefore, a numerical approach of the sponge layer proposed by Mani [
Figure
Flowchart.
A drag coefficient around a sphere is compared between the current Cartesian and BFC solvers. All computational conditions are summarized in Table
Test cases.
Re  Uniform flow  Mesh size  Scheme  Case 

300  0.3  0.10 
A  M03D010A 
0.05 
M03D005A  
0.10 
B  M03D010B  
0.05 
M03D005B  
1.2  0.10 
A  M12D010A  
0.05 
M12D005A  
0.10 
B  M12D010B  
0.05 
M12D005B 
Boundary layer thickness.
Re 

Mesh size 

300  0.058 
0.10 
0.05 
Computational domain (M03D010).
Sphere geometry in Cartesian mesh with IBM.
0.1
0.05
The BFC solver is based on WENOCU6 method developed by Nonomura et al. [
Instantaneous Mach number distributions at Mach 0.3 are obtained as shown in Figure
Mach number distributions at Mach 0.3.
BFC
M03D010A
M03D010B
M03D005A
M03D005B
In the previous 2D study [
Instantaneous distributions of switching parameter
M03D010A
M03D005A
The wake structure of a sphere at Mach 0.3 is visualized by second invariant of velocity tensors (
Isosurfaces of second invariant of velocity gradient tensor (
BFC
M03D010A
M03D010B
M03D005A
M03D005B
Figure
Time histories of drag and lift coefficients at Mach 0.3.
Figure
Timeaveraged total, pressure, and friction drag coefficients at Mach 0.3.
Total drag coefficient
Pressure drag coefficient
Friction drag coefficient
Flow fields around a sphere are solved at freestream Mach number 1.2 to confirm the validity of the current numerical method for supersonic conditions. In the visualization of Figure
Density distributions at Mach 1.2.
BFC
M12D010A
M12D010B
M12D005A
M12D005B
Instantaneous distributions of switching parameter
M12D010A
M12D005A
Timeaveraged drag coefficients at Mach 1.2 in Figure
Timeaveraged total, pressure, and friction drag coefficients at Mach 1.2.
Total drag coefficient
Pressure drag coefficient
Friction drag coefficient
Consequently, the accuracy of the present flow solver was confirmed by the discrepancy with the drag coefficient from the high accurate BFC flow solver. All results could be categorized into “A” (less than 5 percent discrepancy) and “B” (5–10 percent discrepancy) in Table
Performance of drag prediction.
Case  Performance 

M03D010A  B 
M03D010B  B 
M03D005A  A 
M03D005B  A 
M12D010A  B 
M12D010B  B 
M12D005A  A 
M12D005B  A 
To solve a real gasparticle flow at the rocket launch, we have to confirm the validity of a flow around moving particles. Therefore, next validation is conducted to compare drag coefficient around a fixed sphere and a moving sphere. In applying the sharp interface IBM to the moving boundary problem, the socalled “fresh cell” problem [
Computational conditions are summarized in Table
Test cases.
Re  Velocity  Mesh size  State  Case 

300  0.3  0.10 
Fix  M03D010F 
0.05 
M03D005F  
0.10 
Move  M03D010M  
0.05 
M03D005M  
1.2  0.10 
Fix  M12D010F  
0.05 
M12D005F  
0.10 
Move  M12D010M  
0.05 
M12D005M 
Computational domain for moving sphere.
Figure
Initial density contours around a moving sphere (left: Mach 0.3, right: Mach 1.2).
A developed Cartesian solver underestimated the friction drag and overestimated the pressure drag for the fixed sphere analysis as shown in Figures
Time history of drag coefficient for fixed/moving sphere at Mach 0.3 Relative Velocity.
Time history of drag coefficient for fixed/moving sphere at Mach 1.2 Relative Velocity.
A simple IBM for compressible flows was developed and applied to flows around a stationary and a moving sphere. To solve gasparticle flows consisting of shocks and turbulence, a hybrid scheme was employed with the simple IBM. Moreover, a simple and rapid algorithm for force evaluation was implemented and validated. Obtained drag coefficients around a sphere were compared with the ones from high accurate BFC solver. As a result, we could confirm several features of the present numerical method as follows:
Next issue for the gasparticle flow simulation is to treat the momentum and heat exchange between particles and a flow by coupled simulation.
We have two options for the symmetric central difference parts of (
Discrepancy of drag coefficient.
Mach  Scheme  Mesh 




0.3  A  0.10 
4.3  −27.0  50.4 
0.05 
3.1  −16.5  32.1  
B  0.10 
7.7  −22.5  52.2  
0.05 
4.2  −18.1  37.0  
C  0.10 
4.5  −29.6  54.6  
0.05 
4.9  −15.7  35.2  


1.2  A  0.10 
9.4  −2.5  48.6 
0.05 
3.7  −4.4  30.5  
B  0.10 
10.7  −0.9  48.6  
0.05 
3.7  −4.6  30.8  
C  0.10 
10.5  −2.4  53.1  
0.05 
5.6  −2.3  31.5 
To realize largescale computations by high performance computers, simple and easy programs are preferable from the aspect of extensibility and rapid modification. The aerodynamic force evaluation in IBM with Cartesian mesh is one of the concerns due to the implicit boundary representation. The object boundary is approximated as assemblage of planes in 3D or lines in 2D in computational cells in the present IBM. Although it is straightforward procedure that the aerodynamic force is estimated by using the cut planes, it may become time consuming work especially for a moving boundary problem. Nonomura et al. proposed simple force measurement technique for arbitrary geometry based on the staircase representation. We applied the technique to a fixed and moving object and numerical tests were conducted as shown in Table
Test cases.
Re  State  Algorithm  Velocity  Mesh size  Case 

300  Fix  A  0.3  0.10 
FAM03D010 
0.05 
FAM03D005  
1.2  0.10 
FAM12D010  
0.05 
FAM12D005  
B  0.3  0.10 
FBM03D010  
0.05 
FBM03D005  
1.2  0.10 
FBM12D010  
0.05 
FBM12D005  


300  Move  A  0.3  0.10 
MAM03D010 
0.05 
MAM03D005  
1.2  0.10 
MAM12D010  
0.05 
MAM12D005  
B  0.3  0.10 
MBM03D010  
0.05 
MBM03D005  
1.2  0.10 
MBM12D010  
0.05 
MBM12D005 
From the timeaveraged drag coefficient at Mach 0.3 (Figure
Timeaveraged drag coefficient for fixed/moving sphere at Mach 0.3 Relative Velocity.
Total drag coefficient
Pressure drag coefficient
Friction drag coefficient
Timeaveraged drag coefficient for fixed/moving sphere at Mach 1.2 Relative Velocity.
Total drag coefficient
Pressure drag coefficient
Friction drag coefficient
Computational mesh for highorder boundaryfitted coordinate flow solver.
See Figure
The authors declare no conflict of interests.
Part of the computations in this study was conducted by NEC SX9 supported by HPCI (High Performance Computing Infrastructure) hp140138 of the Research Organization for Information Sciences and Technology.