2.4.3 Simplifications to the Navier-Stokes Equations Navier-Stokes方程的简化[cfd-2-4-3]

In the following, we shall consider three common simplifications to the Navier-Stokes equations (2.19). We shall restrict our attention here to the physical reasoning behind each of the approximations. The equations for the first two simplified forms of the Navier-Stokes equations are provided in the Appendix.

When simulating flows around bodies for high Reynolds numbers (i.e., when the boundary layer is thin with respect to a characteristic dimension), the Navier-Stokes equations (2.19) can be simplified. One necessary condition is that there is no large area of separated boundary layer. It can then be anticipated that

only the gradients of the flow quantities in the normal direction to the surface of the body (\(\eta\)-direction in Fig. 2.4) contribute to the viscous stresses [33], [34]. On the other hand, the gradients in the other coordinate directions (\(\xi\) in Fig. 2.4) are neglected in the evaluation of the shear stress tensor (Eqs. (2.14, 2.15)). We speak here of the so-called Thin Shear Layer (TSL) approximation of the Navier-Stokes equations. The motivation for the TSL modification is that the numerical evaluation of the viscous terms becomes computationally less expensive, but, within the assumptions, the solution remains sufficiently accurate. The TSL approximation can also be justified from a practical point of view. In the case of high Reynolds number flows, the grid has to be very fine in the wall normal direction in order to resolve the boundary layer properly. Because of the limited computer memory and speed, much coarser grid has to be generated in the other directions. This in turn results in significantly lower numerical accuracy of the gradient evaluation compared to the normal direction. The TSL equations are for completeness presented in the Appendix (A.6). Due to the fact that secondary flow (e.g., like in a blade row) cannot be resolved appropriately, the TSL simplification is usually applied only in external aerodynamics.

In cases, where the following three conditions are fulfilled:

• the flow is steady (i.e. \(\partial\vec{W}/\partial t = 0\)),
• the fluid moves predominantly in one main direction (e.g., there must be no boundary layer separation),
• the cross-flow components are negligible,

the governing equations (2.19)) can be simplified to a form called the Parabolised Navier-Stokes (PNS) equations [8], [35]-[37]. The above conditions allow us to set the derivatives of \(u\), \(v\), and \(w\) with respect to the streamwise direction to zero in the viscous stress terms (Eq. (2.15)). Furthermore, the components of the viscous stress tensor \(\overline{\overline{\tau}}\), of the work performed by it (\(\overline{\overline{\tau}}\cdot\vec{v}\)), and of the heat conduction \(k\nabla T\) in the streamwise direction are dropped from the viscous flux vector in Eq. (2.23). The continuity equation, as well as the convective fluxes (Eq. (2.21)) remain unchanged. For details, the reader is referred to the Appendix (A.7). Considering the situation sketched in Fig. 2.5, where the main flow direction coincides with the \(x\) coordinate, it can be shown that the PNS approximation leads to a mixed set of parabolic / elliptic equations. Namely, the momentum equation in the flow direction becomes parabolic together with the energy equation, and hence they can be solved by marching in the \(x\)-direction. The momentum equations in the \(y\)- and in the \(z\)-direction are elliptic and they have to be solved iteratively in each \(x\)-plane. Thus, the main benefit of the PNS approach is in the largely reduced complexity of the flow solution - from a complete 3-D field to a sequence of 2-D problems. A typical application of the parabolised Navier-Stokes equations is the calculation of internal flows in ducts and in pipes, and also the simulation of steady supersonic flows using the space-marching method [38]-[41].

As we have seen, the Navier-Stokes equations describe the behaviour of a viscous fluid. In many instances, it is a valid approximation to neglect the viscous effects completely, like for example for high Reynolds-number flows, where the boundary layer is very thin compared to the dimensions of the body. In such cases, we can simply omit the vector of viscous fluxes, \(\vec{F}_v\), from the Equations (2.19). Thus, we are left with

The remaining terms are given by the same relations (2.20)-(2.22) and Eq. (2.25) as before. This simplified form of the governing equations is called the Euler equations. They describe the pure convection of flow quantities in an inviscid fluid. If the Euler equations are formulated in conservative way (like above), they allow for accurate representation of such important phenomena like shocks, expansion waves and vortices over delta wings (with sharp leading edges). Furthermore, the Euler equations served in the past - and still do - as the basis for the development of discretisation methods and boundary conditions.

However, it should be noted that today, due to the computational power of even personal computers and due to the increased demands on the quality of the simulations, the Euler equations are only relatively seldom employed for flow computations.