CAE-Companion-2018-2019

Theory WISSEN CAE

Introduction and Examples of Multiphysics Simulation

Over the last decade, the multiphysics simulation approach replaced the artificial segregation of different physics with a single, unified simulation environment that replicates the real behavior of natural systems. This allowed engineers to simulate the way physics influence one another in the real world in a matter of minutes, drastically reducing the risk of product failure and delays to market. Multiphysics is based on the design to simulate coupled physics effects by solving its underlying mathematical representation based on partial differential equations (PDEs). The user interface should allow the user to include just about any physics effects of interest that are relevant to a specific application, allowing the user to set up a simulation in minutes. For all common multiphysics problems the coupling between the physics involved is fully automated. Joule heating, thermal stress, electrochemical reactions, fluid-struc- ture interaction (FSI) are but a few examples of the many predefined couplings that are available in software packages such as COMSOL Multiphysics. The software then automat- ically compiles the system of PDEs, representing predefined physics as well as user-defined physics, and computes a numerical solution to that system. There are a lot of examples where multiphysical simulation comes into play in automotive applications, starting from sound-vibration couplings via couplings of chemical reactions, heat transport and free or porous flow as e.g. in catalytic con- verters or batteries and fuel cells, to thermal management simulations when designing the electronic system in the car. Sometimes the coupling is just one-directional, where one physics influences the other, sometimes it is bidirectional, where both physical processes influence each other. True multiphysical: The Thermoacoustic Effect The thermoacoustic effect is a truly multiphysical phenome- non as it describes the interaction between acoustic pressure, density and temperature variations. When sound propagates in structures and geometries with small dimensions, the sound waves become attenuated because of thermal and viscous losses. More specifically, the losses occur in the acoustic thermal and viscous boundary layers near the walls. This is a known phenomenon that needs to be included when studying and simulating systems affected by these losses in order to model these systems correctly and to match measurements. The example which is shown here takes this effect into ac- count while modeling an acoustic muffler with perforates. It also shows that this multiphysical approach can be used just for those parts of the model where it plays a significant role while in other parts, modeling the acoustic pressure varia- tions is sufficient to adequately represent real conditions. This

point leads to a second way of Multiphysics coupling: The multiscale-coupling by combining a full-scale model of the systemwith a detailed sub-model of a cutout of the system. Theoretical background of Thermoacoustics For many applications simulating acoustics, a series of assumptions are then made to simplify these equations: the system is assumed lossless and isentropic (adiabatic and reversible). Yet, if you retain both the viscous and heat conduction effects, you will end up with the equations for thermoacoustics that solve for the acoustic perturbations in pressure, velocity, and temperature. The governing equations used in this model are the continu- ity equation:   u    0   i where ρ 0 is the background density; the momentum equation:                    u I I u u u     3 2 0 B T p i where μ is the dynamic viscosity and μ B is the bulk viscosity, and the term on the right hand side represents the diver- gence of the stress tensor; the energy conservation equation:     k T Q i C T T p p       0 0 0    where C p is the heat capacity at constant pressure, k is the thermal conductivity, α 0 is the coefficient of thermal expan- sion (isobaric), and Q is a possible heat source; and finally, the linearized equation of state relating variations in pressure, temperature, and density:   p T T 0 0       where β T is the isothermal compressibility. In thermoacoustics, the background fluid is assumed to be quiescent so that u 0 =0. The background pressure p 0 and background temperature T 0 need to be specified and can be functions of space. The left-hand sides of the governing equations represent the conserved quantities: mass, momentum, and energy (actually entropy). In the frequency domain, multiplication with iω corresponds to differentiation with respect to time. The terms on the right-hand sides represent the processes that locally change or modify the respective conserved quantity. In two of the equations, diffusive loss terms are present, due to viscous shear and thermal conduction. Viscous losses are present when there are gradients in the velocity field, while thermal losses are present when there are gradients in the temperature. Both is usually the case close to solid boundar- ies, where so-called viscous and thermal boundary layers are created at the solid surfaces.         

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