3.4 Initial and Boundary Conditions 初始与边界条件[cfd-3-4]

Regardless of the numerical methodology chosen to solve the governing equations (2.19), we have to specify suitable initial and boundary conditions. The initial conditions determine the state of the fluid at the time \(t=0\), or at the first step of an iterative scheme. Clearly, the better (the closer to the solution) the initial guess will be, the faster the final solution will be obtained. Moreover, the probability of breakdown of the numerical solution process will be reduced correspondingly. Therefore, it is important that the initial solution satisfies at least the governing equations and the additional thermodynamic relations. A common practice in external aerodynamics consists of prescribing freestream values of pressure, density and velocity components (given as Mach number, angle of attack and sideslip angle) in the whole flow field. In turbomachinery, it is important to specify the flow directions in the complete domain to one's best knowledge. The same holds also for the pressure field. It is therefore quite worthwhile to employ lower-order approximations (like potential methods) to generate a physically meaningful initial guess.

Any numerical flow simulation considers only a certain part of the physical domain. The truncation of the computational domain creates artificial boundaries, where values of the physical quantities have to be specified. Examples are the farfield boundary in external aerodynamics; the inlet, outlet and the periodic boundary in the case of internal flows; and finally the symmetry plane. The main problem when constructing such boundary conditions is of course that the solution on the truncated domain should stay as close as possible to a solution which would be obtained for the whole physical domain. In the case of the farfield, inlet and outlet boundaries, characteristic boundary conditions [233]-[235] are often used in order to suppress the generation of non-physical disturbances in the flow field. But despite this, the farfield or the inlet and outlet boundaries may still not be placed too close to the object under consideration (wing, blade, etc.). Otherwise, the accuracy of the solution would be reduced. For external flows, when a lifting body is considered, it is possible to correct the flow variables at the farfield boundary using a single vortex centred at the airfoil or the wing [235]-[237]. In this way, the distance between the body and the farfield boundary can be significantly reduced without impairing the solution accuracy, or improving the accuracy for a given outer boundary position [160], [237], [238]. For internal flow problems, formulations for the inlet and outlet boundaries based on linearised Euler equations and Fourier series expansion of the perturbations were developed [239]-[241]. These formulations allow for a very close placement of the inlet and outlet boundaries to a blade without influencing the solution.

A different type of boundary condition is found when the surface of a body is exposed to the fluid. In the case of inviscid flow governed by the Euler equations (2.45), the appropriate boundary condition is to require the flow to be tangential to the surface, i.e.,

By contrast, for the Navier-Stokes equations no relative velocity between the surface and the fluid immediately at the surface is assumed - the so-called no-slip boundary condition

The treatment of walls becomes more involved in cases, where, e.g., a specified wall temperature distribution has to be met, or when the heat radiation has to be taken into account (see, e.g., [242], [243]).

Furthermore, boundary conditions have to be defined for surfaces where different fluids (e.g., air and water) meet together [244]-[247]. But apart from the physical boundary conditions and those imposed by truncating the flow domain, there can be boundaries generated by the numerical solution method itself. These are for example coordinate cuts and block or zonal boundaries [20]-[26].

The correct implementation of boundary condition is the crucial point of every flow solver. Not only the accuracy of the solution depends strongly on a proper physical and numerical treatment of the boundaries, but also the robustness and the convergence speed are considerably influenced. More details of various important boundary conditions are presented in Chapter 8.