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where dmax is the maximum spacing on the unperturbed orthogonal mesh, A is the waviness amplitude parameter (set to 3), and nj. is the number of half-cycles of the wave along the domain length (set to 9). where dmin is the minimum spacing which occurs near the origin (set to 8 x 10-3 Dey), and s is the geometric grid stretching ratio (set to 1.005). Once a grid is generated for the required domain length (say, up to j = n), it is mirrored first across the x, and then across the y axis. In the next step, this Cartesian orthogonal grid of dimensions (2n+ 1) x (2n+1) is perturbed according to the following (note that the origin now has indices {n+1,n+1} on the unperturbed mesh):

Figure 6 where dmax is the maximum spacing on the unperturbed orthogonal mesh, A is the waviness amplitude parameter (set to 3), and nj. is the number of half-cycles of the wave along the domain length (set to 9). where dmin is the minimum spacing which occurs near the origin (set to 8 x 10-3 Dey), and s is the geometric grid stretching ratio (set to 1.005). Once a grid is generated for the required domain length (say, up to j = n), it is mirrored first across the x, and then across the y axis. In the next step, this Cartesian orthogonal grid of dimensions (2n+ 1) x (2n+1) is perturbed according to the following (note that the origin now has indices {n+1,n+1} on the unperturbed mesh):

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After identification of fluid/solid/surface points is complete, identification of ghost points is performed. A ghost point is any solid point that has exactly one fluid point as a nearest neighbor in a given direction. Note that a solid point can be a ghost point in more than one directions. Consider figure 1, which illustrates a background grid and an IB (the background grid is very coarse to allow better visualization of different aspects of the IBM). The point GP, is a ghost point only in 7 direction and is a solid point in € direction, whereas point GP, is a ghost point in both € and 7 directions. A solid point that has two fluid points as its nearest neighbors in a single direction is marked as a “bad” point in the preprocessing stage. Such a point indicates insufficient resolution of the IB by the background grid. The background grid has to be refined in order to avoid this situation. An example of a bad point in 7 direction is included in figure 1.
After identification of fluid/solid/surface points is complete, identification of ghost points is performed. A ghost point is any solid point that has exactly one fluid point as a nearest neighbor in a given direction. Note that a solid point can be a ghost point in more than one directions. Consider figure 1, which illustrates a background grid and an IB (the background grid is very coarse to allow better visualization of different aspects of the IBM). The point GP, is a ghost point only in 7 direction and is a solid point in € direction, whereas point GP, is a ghost point in both € and 7 directions. A solid point that has two fluid points as its nearest neighbors in a single direction is marked as a “bad” point in the preprocessing stage. Such a point indicates insufficient resolution of the IB by the background grid. The background grid has to be refined in order to avoid this situation. An example of a bad point in 7 direction is included in figure 1.
Figure 2: Image point inside a background grid hexahedron. Tri-linear interpolation needs to be performed on the uniformly-spaced transformed Cartesian grid, on which the governing equations are solved (equations 2 and 3). For this, the coordinates of an image point in the transformed grid must be determined. It is assumed that along a given direction within a given hexahedron, the effective transformed coordinates (denoted by a superscript breve) vary between 0 and 1. Different contributing corner points are then labeled as shown in figure 2. Following the approach of Ghias et al.,!” the closest corner to an image point is first identified. If the closest corner is denoted by CB,, a 2nd- order accurate estimate for effective coordinates of the image point within the corresponding transformed grid hexahedron is given by
Figure 2: Image point inside a background grid hexahedron. Tri-linear interpolation needs to be performed on the uniformly-spaced transformed Cartesian grid, on which the governing equations are solved (equations 2 and 3). For this, the coordinates of an image point in the transformed grid must be determined. It is assumed that along a given direction within a given hexahedron, the effective transformed coordinates (denoted by a superscript breve) vary between 0 and 1. Different contributing corner points are then labeled as shown in figure 2. Following the approach of Ghias et al.,!” the closest corner to an image point is first identified. If the closest corner is denoted by CB,, a 2nd- order accurate estimate for effective coordinates of the image point within the corresponding transformed grid hexahedron is given by
Tere, {%, yc} is the initial vortex center location (set to {0,0}), V; is the vortex strength parameter (set to ).5), re is the vortex core radius (set to 0.8L,), and r = \/(a — a)? + (y — yc)”. For a simulation involving . wall along y = Ywau, initial conditions are specified using both the actual vortex and an image vortex ocated at {Xe image » Ye,image} = {Le , 2 Ywall — Yc}. For the image vortex, the strength parameter is given by /; image = —Vs for the slip-wall boundary condition to be satisfied at y = ywan. In the current simulations, Iwall is set to —2.5739 L,., in order to avoid the wall being coincident with a streamwise grid line in any of the This test involves solution of the 2-D Euler equations (equation 3, with S = 0) for the convection of an inviscid vortex in a uniform freestream at M, = 0.5. In order to study the effect of IBM on the spatial order of accuracy of the base solver, two different simulations are performed: the first involving a vortex convecting only with the freestream, and the second in which a vortex convects with the freestream above a slip-wall (see figure 3). A square-shaped domain extending from —6 L,. to 6 L, in both the x and y directions is used. In the z direction, flow periodicity is assumed in order to allow the vortex to get convected for larger distances using a relatively smaller domain. Following Visbal and Gaitonde,®® the non-dimensional initial conditions for the vortex convecting with the freestream (aligned along the +z axis) are imposed as follows:
Tere, {%, yc} is the initial vortex center location (set to {0,0}), V; is the vortex strength parameter (set to ).5), re is the vortex core radius (set to 0.8L,), and r = \/(a — a)? + (y — yc)”. For a simulation involving . wall along y = Ywau, initial conditions are specified using both the actual vortex and an image vortex ocated at {Xe image » Ye,image} = {Le , 2 Ywall — Yc}. For the image vortex, the strength parameter is given by /; image = —Vs for the slip-wall boundary condition to be satisfied at y = ywan. In the current simulations, Iwall is set to —2.5739 L,., in order to avoid the wall being coincident with a streamwise grid line in any of the This test involves solution of the 2-D Euler equations (equation 3, with S = 0) for the convection of an inviscid vortex in a uniform freestream at M, = 0.5. In order to study the effect of IBM on the spatial order of accuracy of the base solver, two different simulations are performed: the first involving a vortex convecting only with the freestream, and the second in which a vortex convects with the freestream above a slip-wall (see figure 3). A square-shaped domain extending from —6 L,. to 6 L, in both the x and y directions is used. In the z direction, flow periodicity is assumed in order to allow the vortex to get convected for larger distances using a relatively smaller domain. Following Visbal and Gaitonde,®® the non-dimensional initial conditions for the vortex convecting with the freestream (aligned along the +z axis) are imposed as follows:
Table 1: Grid details for the convection of inviscid vortex tests. The simulations are begun with the vortex center located at {0,0} and time-advanced for a period of 480 L,./c;. During this period, the vortex convects through a total distance of 240 L,, and returns to its original location on the actual grid after completing 20 cycles of the periodic streamwise domain. When a slip-wall is employed, the vortex convects with an effective velocity that includes both the freestream and the induced velocity due to the image vortex. However, in the current problem setup, this image-induced velocity magnitude is very low (of the order of 10~° ¢, at the actual vortex center), and hence the final vortex center location is very close to the starting location for simulations involving a slip-wall as well.
Table 1: Grid details for the convection of inviscid vortex tests. The simulations are begun with the vortex center located at {0,0} and time-advanced for a period of 480 L,./c;. During this period, the vortex convects through a total distance of 240 L,, and returns to its original location on the actual grid after completing 20 cycles of the periodic streamwise domain. When a slip-wall is employed, the vortex convects with an effective velocity that includes both the freestream and the induced velocity due to the image vortex. However, in the current problem setup, this image-induced velocity magnitude is very low (of the order of 10~° ¢, at the actual vortex center), and hence the final vortex center location is very close to the starting location for simulations involving a slip-wall as well.
Figure 3: Contours of u/c, and streamline patterns at the end of the simulation on the finest gr ‘Ax = Ay = 0.02 L,.) for the inviscid vortex convection tests.
Figure 3: Contours of u/c, and streamline patterns at the end of the simulation on the finest gr ‘Ax = Ay = 0.02 L,.) for the inviscid vortex convection tests.
Figure 4: Comparison of spatial order of accuracy observed with different configurations for the inviscid vortex convection tests along y axis (a = 0). Dashed lines are 2nd-order least-squares fits to the corresponding data.
Figure 4: Comparison of spatial order of accuracy observed with different configurations for the inviscid vortex convection tests along y axis (a = 0). Dashed lines are 2nd-order least-squares fits to the corresponding data.
Figure 5: A portion of the curvilinear background mesh along with the immersed cylinder boundary used for acoustic scattering by a circular solid cylinder tests. Every 10th point of the background mesh is shown. parameter (set to 3), and Naz; 1s the number of hali-cycles of the wave along the domain length (set to 9) The grid thus generated is presented in figure 5. The maximum domain extents are about —6.14 Dey) 6.14 Dey in both the x and y directions. The planar grid has 627 x 627 grid points. The grid spacing about 0.008 D.y) at the origin, and it increases up to about 0.039 Dy at the boundaries of the domain. Sin the wavelength of the waves generated by the source is 0.25 Deyi, the grid points per wavelength paramet (PPW) varies from about 31.2 at the origin to about 6.4 at the end of the domain. The minimum spat resolution requirements for a reasonable solution to this problem have been found to be about 4 to 6 PP when using the 6th-order compact FD scheme.*® The ability to stretch the background meshes allows t use of more PPW near an immersed boundary, where lower-order explicit FD schemes are used. At t outer boundaries of the x-y domain, a far-field radiation boundary condition?®»?° is used. A time-step At = 5 x 107~°Deyi/cr is used, producing a CFL number of about 0.625. The spatial filtering in the x a: y directions is employed at the end of every 2 complete time-steps. A high value of the filtering paramet (af = 0.49) is used, which was found to be sufficient to maintain numerical stability.
Figure 5: A portion of the curvilinear background mesh along with the immersed cylinder boundary used for acoustic scattering by a circular solid cylinder tests. Every 10th point of the background mesh is shown. parameter (set to 3), and Naz; 1s the number of hali-cycles of the wave along the domain length (set to 9) The grid thus generated is presented in figure 5. The maximum domain extents are about —6.14 Dey) 6.14 Dey in both the x and y directions. The planar grid has 627 x 627 grid points. The grid spacing about 0.008 D.y) at the origin, and it increases up to about 0.039 Dy at the boundaries of the domain. Sin the wavelength of the waves generated by the source is 0.25 Deyi, the grid points per wavelength paramet (PPW) varies from about 31.2 at the origin to about 6.4 at the end of the domain. The minimum spat resolution requirements for a reasonable solution to this problem have been found to be about 4 to 6 PP when using the 6th-order compact FD scheme.*® The ability to stretch the background meshes allows t use of more PPW near an immersed boundary, where lower-order explicit FD schemes are used. At t outer boundaries of the x-y domain, a far-field radiation boundary condition?®»?° is used. A time-step At = 5 x 107~°Deyi/cr is used, producing a CFL number of about 0.625. The spatial filtering in the x a: y directions is employed at the end of every 2 complete time-steps. A high value of the filtering paramet (af = 0.49) is used, which was found to be sufficient to maintain numerical stability.
Figure 6: Contours of instantaneous p’/(p,c?) for scattering of a periodic acoustic source over a circular cylinder test. monitored. After a time interval of about 20 Deyi/c,, asymptotic periodic solution was established at all of the locations. Figure 6 displays the interference pattern generated due to the scattering of sound by the immersed cylinder at a particular instance in time. In order to collect statistical samples for comparison with the analytical solution, a time interval of 40 to 125 D.y/c, is considered.
Figure 6: Contours of instantaneous p’/(p,c?) for scattering of a periodic acoustic source over a circular cylinder test. monitored. After a time interval of about 20 Deyi/c,, asymptotic periodic solution was established at all of the locations. Figure 6 displays the interference pattern generated due to the scattering of sound by the immersed cylinder at a particular instance in time. In order to collect statistical samples for comparison with the analytical solution, a time interval of 40 to 125 D.y/c, is considered.
Figure 8: Comparison of p/,,,/(prc?) along an arc at r/D yi = 5 for scattering of a periodic acoustic source over a circular cylinder test. Figure 7: Comparison of p’,,,,/(prc?) along an arc at r/D.y = 0.55 for scattering of a periodic acoustic source over a circular cylinder test.
Figure 8: Comparison of p/,,,/(prc?) along an arc at r/D yi = 5 for scattering of a periodic acoustic source over a circular cylinder test. Figure 7: Comparison of p’,,,,/(prc?) along an arc at r/D.y = 0.55 for scattering of a periodic acoustic source over a circular cylinder test.
Figure 9: Comparison of p',,,</(prc2) along a ray at 6 = 7/2 for scattering of a periodic acoustic source over a circular cylinder test.
Figure 9: Comparison of p',,,</(prc2) along a ray at 6 = 7/2 for scattering of a periodic acoustic source over a circular cylinder test.
where {2,, yc} is the initial pulse center location (set to {4 D.yi, 0}), b is the source width parameter (set to 0.2 Dey). M, and M, are the freestream Mach numbers in the x and y directions, respectively. Both are set to zero. The perturbation scaling factor (€) is set to 10~?. The analytical solution to this problem is available is ref. 43. quation 3, with S = 0) for this problem, the non-dimensional flow field is initialized with the following
where {2,, yc} is the initial pulse center location (set to {4 D.yi, 0}), b is the source width parameter (set to 0.2 Dey). M, and M, are the freestream Mach numbers in the x and y directions, respectively. Both are set to zero. The perturbation scaling factor (€) is set to 10~?. The analytical solution to this problem is available is ref. 43. quation 3, with S = 0) for this problem, the non-dimensional flow field is initialized with the following
isolate the effectiveness of IBM in scattering an incident acoustic wave. The excellent agreement observed for this problem on the wavy mesh is encouraging. Figure 10: Contours of p’/(p,c?) at several instances in time for scattering of an initial Gaussian pulse over a circular cylinder test.
isolate the effectiveness of IBM in scattering an incident acoustic wave. The excellent agreement observed for this problem on the wavy mesh is encouraging. Figure 10: Contours of p’/(p,c?) at several instances in time for scattering of an initial Gaussian pulse over a circular cylinder test.
For all the simulations presented in this section, a single Cartesian stretched grid is utilized. The computational domain is a rectangular parallelepiped extending from %pin = —4 Dsphr tO Tmax = 14 Dspnr in the streamwise direction, and from —4 Dspnr to 4 Dspnr in the lateral (y and z) directions. A triangulated spherical surface of diameter D,,,, and centered at the origin is immersed in the background grid. A portion of the background mesh in the vicinity of the immersed sphere is presented in figure 12. A Cartesian grid results in a very inefficient grid point distribution for this problem; however, it is used here to demonstrate the generality of the method. The grid has Nz x Ny x Nz = 382 x 297 x 297 points. Grid stretching is used to cluster more grid points towards the immersed sphere. The smallest background grid spacing neat the sphere surface is 5 x 1073 Dspnr- This is equal to the smallest spacing used in the numerical study of this problem by Johnson and Patel.*° The laminar axisymmetric boundary layer thickness at 6 = 90° from the front stagnation point on a sphere can be approximated by 6/Dspnr = 1.64/ VRe,*® where 6/ Dsphr is defined as the distance away from the surface where the actual local velocity magnitude is 99% of the local velocity magnitude predicted by the potential flow theory. For the largest Reynolds number discussed in this section (Re = 300), the background grid has about 16 grid points within the boundary layer at 6 = 90°. The largest background grid spacing near the sphere surface is 1.9 x 107? Dgppr, which occurs in the streamwise direction at 9 = 90°. The grid stretching ratio is limited to about 2% up to = 5 Dsphr in the & direction. and to about 3% up to 2 Dspnr in ty and +z directions. The maximum grid stretching ratio, towards the streamwise exit of the domain, is about 5%. A far-field radiation boundary condition?®:?° is applied at the four lateral boundaries of the domain. At the streamwise inlet and outlet of the domain, characteristic-based inflow and outflow boundary conditions®® are used, respectively. The spatial filtering operation is performed at the end of every complete time-step, and the filtering parameter ay is set to 0.47 in all the simulations presented in this section.
For all the simulations presented in this section, a single Cartesian stretched grid is utilized. The computational domain is a rectangular parallelepiped extending from %pin = —4 Dsphr tO Tmax = 14 Dspnr in the streamwise direction, and from —4 Dspnr to 4 Dspnr in the lateral (y and z) directions. A triangulated spherical surface of diameter D,,,, and centered at the origin is immersed in the background grid. A portion of the background mesh in the vicinity of the immersed sphere is presented in figure 12. A Cartesian grid results in a very inefficient grid point distribution for this problem; however, it is used here to demonstrate the generality of the method. The grid has Nz x Ny x Nz = 382 x 297 x 297 points. Grid stretching is used to cluster more grid points towards the immersed sphere. The smallest background grid spacing neat the sphere surface is 5 x 1073 Dspnr- This is equal to the smallest spacing used in the numerical study of this problem by Johnson and Patel.*° The laminar axisymmetric boundary layer thickness at 6 = 90° from the front stagnation point on a sphere can be approximated by 6/Dspnr = 1.64/ VRe,*® where 6/ Dsphr is defined as the distance away from the surface where the actual local velocity magnitude is 99% of the local velocity magnitude predicted by the potential flow theory. For the largest Reynolds number discussed in this section (Re = 300), the background grid has about 16 grid points within the boundary layer at 6 = 90°. The largest background grid spacing near the sphere surface is 1.9 x 107? Dgppr, which occurs in the streamwise direction at 9 = 90°. The grid stretching ratio is limited to about 2% up to = 5 Dsphr in the & direction. and to about 3% up to 2 Dspnr in ty and +z directions. The maximum grid stretching ratio, towards the streamwise exit of the domain, is about 5%. A far-field radiation boundary condition?®:?° is applied at the four lateral boundaries of the domain. At the streamwise inlet and outlet of the domain, characteristic-based inflow and outflow boundary conditions®® are used, respectively. The spatial filtering operation is performed at the end of every complete time-step, and the filtering parameter ay is set to 0.47 in all the simulations presented in this section.
Table 2: Simulation details for the steady viscous flow over a sphere tests. Figure 13: Axisymmetric steady streamline pattern over the sphere for Re = 200. Various geometrical flow features mentioned in the text are represented qualitatively.
Table 2: Simulation details for the steady viscous flow over a sphere tests. Figure 13: Axisymmetric steady streamline pattern over the sphere for Re = 200. Various geometrical flow features mentioned in the text are represented qualitatively.
Table 3: Comparison between the current and reference results for viscous flow over a sphere at Re = 300
Table 3: Comparison between the current and reference results for viscous flow over a sphere at Re = 300
Figure 15: Extrapolated mean pressure coefficient at different Reynolds numbers over the surface of the immersed sphere, from the front stagnation point (6 = 0°) to the rear end (# = 180°). Solid lines are current results, symbols are reference values from DNS by Tomboulides and Orszag.*”
Figure 15: Extrapolated mean pressure coefficient at different Reynolds numbers over the surface of the immersed sphere, from the front stagnation point (6 = 0°) to the rear end (# = 180°). Solid lines are current results, symbols are reference values from DNS by Tomboulides and Orszag.*”
Figure 16: A view of the vortex structures at an instance during the viscous flow over a sphere test at Re = 300. Iso-surfaces of the Q-criterion are plotted at Q = 0.005 (Us /Dspnr)?-
Figure 16: A view of the vortex structures at an instance during the viscous flow over a sphere test at Re = 300. Iso-surfaces of the Q-criterion are plotted at Q = 0.005 (Us /Dspnr)?-
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