On the Use of the Immersed Boundary Method for Engine Modeling

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1 On the Use of the Immersed Boundary Method for Engine Modeling K. J. Richards, E. Pomraning, P. K. Senecal Convergent Thinking C. Y. Choi Caterpillar Inc. C. J. Rutland University of Wisconsin--Madison Introduction Historically, boundary fitted grids have been used to model complex geometries. The authors are investigating a novel grid generation concept, Immersed Boundary Method, as an improved methodology to boundary fitted grids. There are two significant disadvantages to using a boundary fitted grid. First, whether structured or unstructured, fitting a grid to a complex geometry prevents the use of simple orthogonal grids, thereby foregoing the benefits of numerical accuracy and computational efficiency associated with simple orthogonal grids. Second, generating a boundary fitted structured grid for a complex geometry can be extremely time consuming and difficult. Often, the grid generation difficulties and time can be a roadblock in simulating complex geometries such as an internal combustion engine. These problems can be overcome by using the immersed boundary method. The immersed boundary method models complex geometries by representing the necessary shape through forces within the grid. This concept was first introduced by Peskin (1972) for performing simulations of flow in a heart. The work of Peskin allows for a two-way coupled treatment of the immersed boundary. However, this method has severe drawbacks in terms of CFL requirements for stability. Other researchers (Mohd-Yusof, 1997, Verzicco et al., 2000, Fadlun et al., 2000 ) have improved on Peskin s technique for the specific case of a solid wall where there is no need to couple the effect of the fluid to the wall treatment. By using the immersed boundary approach, the underlying grid need not coincide with the geometry being solved. Since the grid does not need to coincide with the surface geometry, the type of grid used is chosen for computational efficiency instead of geometry, which allows the use of simple orthogonal grids. Furthermore, the grid generation complexity and time is greatly reduced, as the complex geometry only needs to be mapped onto the underlying orthogonal grid. Currently, Convergent Thinking in cooperation with the University of Massachusetts Amherst is developing the immersed boundary code MOSES (MOdifiable Source Engine Simulation) for use in engine simulations. In this paper, we present immersed boundary background, improvements to the immersed boundary method, progress in the development of the code, and MOSES simulation results. Immersed Boundary Background The dynamics of fluid flow are governed by equations that describe the conservation of mass, momentum, and energy, typically referred to as the Navier-Stokes equations (see e.g., Warsi, 1993). These equations can also include additional transport equations for scalars of interest. The fully compressible 3-d equations are solved in MOSES. For brevity, the method is

2 only shown for the momentum equation; however, it should be clear how to apply the method to other conservation equations. The compressible momentum equation is give by ρu ρuu 2 i i j P u u i j u + = + µ + µ k ij Fi t x 3 j xi xj xj xi x δ + (1) k where the term F i is the forcing function. For cases where the immersed boundary represents a solid, Mohd-Yusof proposed a method for deriving the necessary forces to represent the surface (Mohd-Yusof, 1997). The forcing concept is based on a one-way coupling between the solid and the fluid i.e., the solid influences the fluid, but the fluid does not influence the solid. This type of formulation works well in a case like a reciprocating engine where the motion of the piston can be prescribed independent of the fluid behavior (Verzicco et al., 2000). For purposes of describing this method, a first-order time difference is used. The method does not require the use of first-order temporal differencing it is simply used here for clarity in describing the method. The forcing can be represented using a first order time difference as shown in Equation (2) (Fadlun et al., 2000). n 1 ( + n ui ui ) ρ = RHSi + F i (2) dt Note that the Right Hand Side (RHS) is composed of the convective, diffusive, and pressure terms in the momentum conservation equation. The term F i represents a force in the momentum conservation equation. The Mohd-Yusof method prescribes the value of the velocity for the fluid nodes immediately next to the immersed boundary. For direct forcing, the value of the velocity needs to be known for the nodes where the velocity is to be prescribed; how this velocity is known will be described later. With the velocity known, the force is simply n ( vb ui ) Fi = RHSi + ρ (3) dt where v b is the velocity to be imposed on the node of interest (Fadlun et al., 2000). Notice that when this force is applied to Equation (1), the net result is to cancel terms in the equation, unlike the method of Peskin where new terms arise (McQueen and Peskin, 1989). Because of this nature of the direct forcing, the stability of the solver is not affected, making the overall CFL constraint unaffected by the immersed boundary. Verzicco et al. (2000) have developed a 2 nd order accurate method for determining the boundary velocity. This method uses the value of the velocity at the wall along with the second fluid node in the domain to determine the value of the force at the first fluid node. The second fluid node is solved as an interior fluid node, and then this solved value is used to do a linear interpolation between it and the boundary to find the velocity at the desired node. Figure 1 gives an example of how this is done. The velocity at the node next to the immersed boundary is not solved for by use of the Navier-Stokes equations it is simply an interpolation between two known values. Note that when using an implicit solver, this value must be updated at each iteration step because there is coupling between its value and the values that will be found in the interior.

3 Figure 1: method used for determining an appropriate boundary force (Fadlun, et al., 2000). (Note: the boundary is indicated by the curved line and is assumed to be moving upwards at a fixed velocity, V) MOSES Code MOSES is a 3-dimensional finite difference compressible Navier-Stokes code written in the C programming language that uses the PISO method (Issa, 1984, 1985) to solve the conservation equations. The gas-phase solver, combustion/chemistry solver, and the immersed boundaries are being developed at Convergent Thinking while the spray models are being developed at the University of Massachusetts Amherst. The code is arbitrary order in space (up to 20 th order) and 1 st or 2 nd order accurate in time. For turbulence modeling, the k-ε and RNG k- ε models have been implemented into the code while for combustion, a detailed chemistry solver developed at Convergent Thinking called SAGE (Senecal et al., 2003) is currently being implemented into the code. At this time, inflow and outflow conditions have been implemented in the code and are handled using the immersed boundary concept. Moving boundaries are currently not possible but work is underway on implementing them. Immersed Boundary Implementation One issue inherent in the use of the Immersed Boundary Method is the ability of pressure to see through the boundary. This problem is even more evident when a boundary exists between fluid nodes the pressure solver does not see the existence of a boundary between the nodes because they are logically connected. In MOSES, this problem is alleviated by the use of a private stencil. When a node is next to a wall, additional artificial nodes are created to complete the derivative stencils needed by the fluid node. Figure 2 illustrates this concept. For a simulation using 4 th order central derivatives, the black node next to the wall needs two nodes inside the solid to complete its stencil. The gray nodes in the solid are created and are only seen by the black node. The values for pressure, temperature, and velocity for these artificial nodes can be manipulated by the solver to ensure that the node next to the wall sees the appropriate boundary condition. The method also works for the case of an infinitely thin boundary between fluid nodes. The nodes on either side of the boundary would see their own private nodes rather than seeing the fluid nodes on the opposite side of the thin wall.

4 fluid solid y x Figure 2: Private stencil concept. When solving for the immersed boundary velocity, there are cases where a boundary node will see the boundary in two (or more) different directions as seen in Figure 3. This case is handled by calculating an appropriate boundary condition for the two (or more) directions and then performing an inverse distance weighted average for all necessary directions given by u u x y + 1 u () = 2 i i (4) where u x the x-direction. 1 2 is the velocity boundary condition independently solved for using the boundary seen in solid 2 fluid 1 Figure 3: Depiction of a scenario where a fluid node sees two boundaries. Results To show the capability of the immersed boundary method, an engine intake simulation is presented as an example. The relevant simulation parameters are shown in Table 1. As stated earlier, grid generation using the immersed boundary method is quick and simple. The grid is generated by simply immersing the CAD surface (e.g., STL) into the underlying orthogonal grid as can be seen in Figure 4. The grid resulting from this process is also shown in Figure 4. The boundary conditions, run parameters, and grid data are stored in text input files that are read in by MOSES. The results of the MOSES simulation are shown in Figures 5 and 6. Since no turbulence model is used (stability is maintained by upwinding) the results are unsteady LES like flows.

5 Table 1: Engine specifications, boundary conditions, and numerical parameters Parameter Engine bore Displacement Inlet temperature Inlet total pressure Outlet temperature Outlet static pressure Wall temperature Fluid Wall boundary condition Turbulence model Numerical accuracy Value cm 2.44 L 400 K Pa 300 K Pa 300 K Air No-Slip None 2nd Order Time and 3rd Order Space (a) (b) Figure 4: STL file immersed in orthogonal grid (a), and resulting grid (b). (a) (b) Figure 5 Side view of temperature field in K (a), and side view of velocity field in m/s (b).

6 (a) (b) Figure 6 Top view of velocity field in m/s just below valves (a), and top view of velocity in m/s middle of cylinder (b). References Fadlun, E.A., Verzicco, R., Orlandi, P., and Mohd-Yusof, J., Combined Immersed-Boundary Finite- Difference Methods for Three-Dimensional Complex Flows Simulations, Journal of Computational Physics, Vol. 161, Issa, R.I., Solution of the Implicitly Discretised Fluid Flow Equations by Operator-Splitting, Journal of Computational Physics, Vol. 62, Issa, R.I., Gosman, A.D., and Watkins, A.P., The Computation of Compressible and Incompressible Recirculating Flows by a Non-iterative Implicit Scheme, Journal of Computational Physics, Vol. 62, McQueen, D.M. and Peskin, C.S., A Three-Dimensional Computational Method for Blood Flow in the Heart. II. Contractile Fibers, Journal of Computational Physics, Vol. 82, Mohd-Yusof, J., Combined Immersed-Boundary/B-Spline Methods for Simulations of Flow in Complex Geometries, CTR Annual Research Briefs, NASA Ames Research Center/Stanford Univ. Center for Turbulence Research, Stanford, CA, 1997 Peskin, C.S., Flow Patterns Around Heart Valves: A Numerical Method, Journal of Computational Physics, Vol. 10, Senecal, P. K., Pomraning, E., Richards, K. J., Briggs, T. E., Choi, C. Y., McDavid, R. M., and Patterson, M. A., Multi-Dimensional Modeling of Direct-Injection Diesel Spary Liquid Length and Flame Lift-off Length using CFD and Parallel Detailed Chemistry, SAE , 2003 Verzicco, R., Mohd-Yusof, J., Orlandi, P., and Haworth, D., Large Eddy Simulation in Complex Geometric Configurations Using Boundary Body Forces, AIAA Journal, Vol. 38, No. 3, Warsi, Z. U. A., Fluid Dynamics: Theoretical and Computational Approaches, CRC Press, 1993.

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