Direct Modeling for Computational Fluid Dynamics Kun Xu February 20, 2013 Computational fluid dynamics (CFD) is new emerging scientific discipline, and targets to simulate fluid motion in different scales. For a physical reality, the description of the flow behavior depends closely on the scales we are trying to identify or see it. Any theoretical equation, such as the Boltzmann equation or the Navier-Stokes equations, is derived based on the physical modeling and is valid in certain scale. In general, the aim of the CFD is not targeting on the numerical solution of any specific partial differential equation, but presenting a corresponding physical flow behavior in the mesh size scale. The principle of the CFD should be based on a direct modeling of the flow motion in a discretized space. Due to the freedom in choosing the mesh size or the resolution we can afford to look at the fluid, there is a large variation of cell size in relative to the intrinsic physical scale of the different level of flow description. The numerical principle for CFD is now becoming a multiscale modeling problem, which is capable to identify the corresponding flow behavior in a numerical cell resolution. Here the multiple scales mean the physical scale of the flow motion and the numerical scale to identify it. Since any governing equation is derived based on its own physical modeling scale, to follow or discretize a single governing equation, such as NS one, will not be a proper way in CFD research. Also, to first construct an appropriate governing equation in the mesh size scale, then to solve it numerically may not be easy. A possible way is to make mesh size fine enough everywhere to resolve the smallest physical scale, such as the local particle mean free path of the Boltzmann equation. But, to have such a fine resolution is not practical due to its computational cost and is not necessary in practical engineering applications. So, instead of solving any specific governing equation, the numerical principle of CFD becomes the direct modeling and simulation of the flow evolution in the mesh size scale. The unified gas-kinetic scheme developed recently provides such a method with direct capturing of continuous flow behavior from microscopic to macroscopic level with a variation of mesh size. 1
1 Physical Modeling of Fluid Motion There are different levels in flow modelings. In the history of fluid mechanics, the traditional theoretical approach is to apply physical laws, such as mass, momentum and energy conservation, in a small control volume, and with the assumption of constitutive relationship in the macroscopic scale or direct particle movement in the microscopic scale, and to construct the connection among all physical variables inside the control volume and surface fluxes. Then, as the control volume shrinking to zero, with the assumption of smoothness of the flow variables in the scale of control volume, corresponding partial differential equations (PDEs) are obtained. For the PDEs, even with the continuous space and time, the validity of the equation is still on its modeling scale, such as the scale for the validity of constitutive relationship, and the scale where dissipative coefficients can be properly defined. For example, for the Boltzmann equation, the modeling scale is the particle mean free path and the mean particle collision time. For the NS equations, the valid scale is the dissipative wave structure. For the Euler equations, the scale is the advection transport wave structure. The partial differential equations constructed so far can be the potential flow, the Euler equations, the Navier-Stokes equations, the Boltzmann equation, or molecular dynamic equation. With the well-defined PDEs, many research work concentrate on the mathematical analysis of these equations, or try to solve them analytically with different level of assumption, so called limiting case solutions. 2 Numerical Computation of the Fluid Motion With the theoretical PDE, the traditional CFD is to discretize the PDE without referring to the underlying scale of the equation. The numerical cell size scale is pretended to be the same scale as the physical modeling scale of the PDE. Certainly, in order to get valid numerical solution, the numerical cell size should be compatible with the modeling scale of thepde.forexample, inordertosolve thekineticequation, acell sizeshouldbeonthesame scale as the particle mean free path. But mostly, in the numerical PDE methodology, the intrinsic modeling scale of the equation has not been taken into account seriously. A direction discretization, such as f x = (f i+1 f i 1 )/2 x, is usually blindly used. When the cell size is not on the the same level as the modeling scale of the equation, inaccurate numerical solution may be obtained. In order to improve the accuracy of the scheme, the numerical analysis is used, such as figuring out the truncation error of the numerical scheme and obtaining the modified equation. In order to reduce the truncation error and get a stable and reliable scheme, the numerical discretization is further modified to reduce the truncation error and improve the order of the scheme. Unfortunately, the flow motion is associated with multiple 2
scale nature with different structure thickness. The truncation error will be a function of flow structures. This kind of numerical PDE methodology is basically to design a scheme from its outcome result. This process has no any influence from fluid physics itself. Due to the lack of guiding principle, and the absence of the physical impact from the numerical cell resolution, to develop a sounding scheme in this way is almost impossible. It is more likely to take a shotting without understanding the flying dynamics of the bullet. Even getting to the target in a special case, the generality and applicable regime of the method are limited. In the past decades, the equations to be solved numerically cover the potential flow (60s-70s), the Euler equations (70s-80s), the NS equations (80s - now), and the Boltzmann type equations (90s- now). Even though these equations have been routinely used in a wide range of engineering applications, the acceptability of the scheme is more or less through the Verification and Validation(V&V) process. Due to the absence of numerical principles in the design of the scheme, there is no predicability of the mechanism of the scheme before testing it. Even with great success in the CFD research in the past decades, at the current stage, we are still facing many problems, such as the numerical shock instability in high Mach number flows and the valid approach for high-order schemes for the capturing both continuous and discontinuous flow evolution. Now we are urgently facing a fundamental question: what is the numerical principle for computational fluid dynamics? For any scientific research, a principle is needed once the subject needs to be promoted from the experiencing study to a scientific discipline. The theoretical fluid mechanics presents modeling equations in continuous space and time. In the computational aspect, the central point is to describe the flow motion in a discretized space. The fluid dynamics study in this space is more challenging, which has to take into account both the physical scale of flow motion and the numerical scale to represent it. In other words, we are working in a space with limited resolution. The principle of CFD is to model the gas evolution directly in a numerical cell size scale, and this principle is different from constructing and discretizing any specific PDE of the flow motion. In order to construct an appropriate physical modeling in a discretized space, we need first to know the intrinsic properties of flow molecules. A real flow is composed of molecules, and there are kinematic and dynamic properties for these molecules, such as molecular mass, mean free path, collision time, and interaction potential between colliding molecules, and averaged density and velocity in a defined scale. The representation of the flow in a numerical scheme depends on the cell resolution. In a discretized space, in order to identify different flow behavior, we need first to define a numerical cell size Knudsen number, i.e., Kn c = λ/ x, which gives a connection between the physical gas property (the mean free path λ ) and the cell resolution ( x). The cell Knudsen number is a parameter to connect the numerical space to the molecular reality. In a flow computation, due to the freedom 3
in choosing the cell size, the cell s Knudsen number can cover a wide range of values. For example, for a high speed space shuttle flying at high altitude, the mesh size can take the value of the molecular mean free path in the non-equilibrium shock regions and hundreds of mean free path close to the surface or far away from the shuttle. At different location, different values of cell Knudsen numbers correspond to different dynamics of flow motion. The CFD research is basically to model the flow motion in the cell size scale. Here we are not considering multiple scales of the physical flow itself, such as diffusive or turbulent mixing scales, because these are phenomenological observation which cannot have a direct connection with the fundamental CFD principle. What we would like to concentrate in a computation is to model the dynamics associated with different order of Kn c. Certainly, we can describe the flow problem in a single scale, such as taking Kn c 1, where the mesh size is on the same order as the particle mean free path everywhere. However from a computational point of view, the cost to use such a mesh is too high. To resolve all these small scales can be prohibitively expensive. The choosing of the numerical mesh size needs to take into account the balance between accuracy of recovering the physical phenomena and the efficiency to achieve such a solution. Briefly, the main aim of the CFD, is to provide a multi-scale modeling method to be able to capture both microscopic and macroscopic phenomena at different Kn c number regions. In order to design such a multi-scale modeling scheme, the flow evolution with different Kn c numbershastobecaptured. IntheregionsofKn c 1,thesamemodelingmechanismof deriving the Boltzmann equation needs to be used numerically to design such a scheme, where the particle transport and collision can be treated separately. In the region of Kn c 1, the modeling in deriving the NS equations should be recovered in the numerical process. For a even large Kn c value, the cell size may be compatible with the whole domain of the flow motion, a simple averaging of the flow variables inside such a domain may be the modeling physics we can provide. In the above modeling for different Kn c, there is no any fixed equation to be used. The direct adaptation of different PDE in different regions will be associated with great difficulties, if not impossible. Many physical problems do need a unified approach to have a valid description in all Kn c regimes. In the aerospace, we need to resolve the non-equilibrium flow region with the cell resolution on the order of particle mean free path and the continuum flow region to resolve the dissipative boundary layer with the cell resolution being much larger than the local particle mean free path. In plasma physics, one needs to recover macroscopic modeling in the quasi-neutral region and kinetic modeling in non-quasineutral region, and there needs a smooth dynamic transition between these models. The another goal for the development of the unified scheme is not to use the domain decomposition to construct and match different models at different scales through an interface or buffer zone. The unified approach does not 4
require the transform of data from one-scale to another, which is often difficult and most of the time has no unique solution. The unified method avoids the coupling of different models. The way to provide a unified approach is to take a direct modeling, where the solution transits smoothly in different regimes. 3 Unified Gas-kinetic Scheme for CFD The methodology of the development of the unified scheme is based on the direct physical modeling in a discretized space. The target here it to figure out what happening in the numerical cell size scale. For example, if the cell size is much less than the particle mean free path, i.e., Kn c 1, the particle free transport will be identified in the numerical cell size scale. When Kn c 1, both free transport and particle collision needs to be taken into account. WhenKn c 1,theindividualparticlemotionisnotimportant. Insuchameshsize scale, the pressure wave due to the accumulation of massive number of particle collisions and collective advection will be modeled and used in the numerical modeling. In order to capture different flow behavior in different cell size, a time evolution solution of the kinetic level PDE will be used, and this solution should cover from the kinetic to the hydrodynamic flow physics in the evolution process. The use of the PDE s time evolution solution for the modeling and the direct numerical discretization of the same PDE are different concepts. Sometimes, we also call the current approach as the PDE-based modeling, instead of numerical PDE. For an analytic time evolution solution, the valid scale of this solution can go beyond the modeling scale of the PDE. For example, the Boltzmann equation itself is valid in the mean collision time scale τ, but its time evolution solution with the initial condition on the scale of cell size can be valid in a much longer time, such as t >> τ, such as the time for the wave propagation across cell size scale. The key of the unified scheme developed recently is to construct a gas evolution process around a cell interface and obtain its time dependent solution. This solution can cover the flow physics from the free molecular motion to the viscous heat conducting wave interaction of the NS level. In the unified scheme, there is great variation of the ratio of time step t to the local particle collision time τ, i.e., the socalled time step Knudsen number Kn t = τ/ t. Which flow physics the numerical modeling represents depends on the local value Kn t. The direct modeling in CFD connects closely the physical flow behavior in a mesh size scale. 5
4 Differences between Multiscale Modeling and Direct Modeling The macroscopic Navier-Stokes equations can be formally derived from the Boltzmann equation, which in turn can be derived from the Liouville equation via the BBGKY hierarchy. Many physical assumptions are built into these derivations and their validations are warranted under certain physical situations. In the current theoretical framework of fluid dynamics, there are only a few distinct governing equations, such as the Boltzmann, Navier-Stokes, and the Euler equations. These equations are constructed to describe flow physics at different scales. Between these equations, we may have also the Burnett, or super-burnett, and many kind of moment equations. But, their validations can be hardly judged from the physical point of view, because they are not directly obtained from first physical principle through physical modeling, rather than derived from mathematical manipulation of the kinetic e- quation and in the hope that the derived equations are valid in another scale. Instead of deriving and using of these distinct macroscopic equations, there is a need to investigate other possible theoretical approach for fluid dynamic study, especially numerically. A direct question we would like to ask is: why are there only distinct governing equations for the fluid modeling? With the continuous change of spatial and temporal scales, Could we have a continuous spectrum of governing equations? The central idea of direct modeling presented is to figure out the flow motion with a continuous spectrum of governing equations numerically. The current CFD methods are mostly based on the direct numerical discretization of the macroscopic governing equations. As a consequence, the numerical schemes for macroscopic equations are very efficient but are often not accurate enough for non-equilibrium flow computations. Microscopic equations, on the other hand, may have better accuracy, but they are usually too expensive to be used to model systems in a large scale. The current mutliscale modeling is basically to construct valid technique which combines the efficiency of macroscale models and the accuracy of microscale models. The accuracy of any specific constitutive relationship in the macroscopic equations is often questionable. In many cases, the constitutive relations become quite complicated, where too many parameters need to be fitted, the physical meaning of these parameters becomes obscure. Consequently the original appeal of simplicity and universality of macroscopic model is lost. Central to the multisale coarse-grained modeling is the construction of the constitutive relation which represents the effect of the microscopic processes at the macroscopic level. This strategy of multiscale modeling is different from the direct modeling methodology presented in the unified scheme. There is no combination of the macro and micro description in the direct modeling. Any observation in a certain scale is the accumulating and evolving solution from the microscopic model through a physical coarse grained averaging. Furthermore, in the direct modeling, 6
with the variation of the observation scale or the mesh size scale, a continuous spectrum of governing equations are basically recovered computationally, and it is not necessary to write down their formulations explicitly. It is not needed to know the corresponding micro-scale and macro-scale interactions and the constitutive relationship. With the variation of scale, there is no longer any distinct micro and macro flow physics definition. 5 The Necessity of Direct Modeling for CFD The direct modeling is not just to develop numerical algorithms, in certain sense it provides a continuous spectrum of physical modeling at different scales. The accuracy of the direct modeling depends on the accuracy of the finest fluid dynamic description. Currently, we will only resolve the flow evolution solution up to the mean free path scale, therefore the Boltzmann equation will be used as the finest fluid dynamic modeling. Since the evolution solution of the Boltzmann equation will be used in different scales, along with the coarse grained averaging effect on a large scale, the corresponding coarse level gas evolution model will be obtained. Therefore, the direct modeling principle is not intended to solve the Boltzmann equation, but use its solution to do the modeling. In the macroscopic level, the fine structure in the kinetic level is lost, and is actually not needed. The NS solution is recoveredinalargescaleandthisrecoveredsolutionisnotsensitivetothekineticmodelatall. In other words, the direct modeling introduces a variation of governing equations, it captures both microscopic and macroscopic flow behavior in the corresponding cell resolution. The limited cell resolution, i.e., the same as the uncertainty theory in the quantum mechanics, effects the observable fluid dynamics. Any subcell structure is associated with uncertainty. Fortunately, in the direct modeling method we don t need to feed back the macroscopic evolution solution back to the microscopic level. The CFD research has made great progress in the past decades. There is no doubt that CFD will become even more important in fluid mechanics study. The traditional approach of direct discretization of PDE, and the independence of the intrinsic physical scale of the governing equations from the numerical cell size representation, limits the further development of the CFD methods. The direct modeling is to present a new CFD principle, which directly connects the fluid evolution process with the numerical mesh resolution, and simulate the fluid dynamic evolution through a continuous variation of scales. 7