2.4 Complete System of the Navier-Stokes Equations 完整的Navier-Stokes方程组[cfd-2-4]

In pure aerodynamics, it is generally reasonable to assume that the working fluid behaves like a calorically perfect gas, for which the equation of state assumes the form [13], [14]

where \(R\) denotes the specific gas constant. The enthalpy results from

It is convenient to express the pressure in terms of the conservative variables. For that purpose, we have to combine Equation (2.12), relating the total enthalpy to the total energy, together with the equation of state (2.26). Substituting expression (2.27) for the enthalpy and using the definitions

we finally obtain for the pressure

The temperature is then calculated with the aid of the relationship Eq. (2.26). The coefficient of the dynamic viscosity \(\mu\) is, for a perfect gas, strongly dependent on temperature but only weakly dependent on pressure. The so-called Sutherland formula is frequently used. The result for air is (in SI units)

where the temperature \(T\) is in degree Kelvin (K). Thus, at \(T = 288\,\mathrm{K}\) one obtains \(\mu = 1.78\cdot 10^{-5}\,\mathrm{kg/ms}\). The temperature dependence of the thermal conductivity coefficient \(k\) resembles that of \(\mu\) in the case of gases. By contrast, \(k\) is virtually constant in the case of liquids. For this reason, the relationship

is generally used for air. In addition, it is commonly assumed that the Prandtl number \(Pr\) is constant in the entire flow field. For air, the Prandtl number takes the value \(Pr = 0.72\).

The matter becomes more complicated when one has to deal with a real gas. The reason is that now we have to model a thermodynamic process and chemical reactions in addition to the fluid dynamics. Examples for a real gas flow are the simulation of combustion, the hypersonic flow past a re-entry vehicle, or the flow in a steam turbine. In principle, two different methods can be pursued to solve the problem. The first methodology is applicable in cases, where the gas is in chemical and in thermodynamical equilibrium. This implies that there is a unique equation of state. Then, the governing equations (2.19) remain unchanged. Only the values of pressure, temperature, viscosity, etc. are interpolated from lookup tables using curve fits [15]-[17]. But in practice, the gas is more often in chemical and/or thermodynamical non-equilibrium and has to be treated correspondingly.

Let us for illustration consider a gas mixture consisting of \(N\) different species. For a finite Damköhler number, which is defined as the ratio of flow-residence time to chemical-reaction time, we have to include finite-rate chemistry into our model. It has to describe the generation/destruction of species due to chemical reactions. In what follows, we will furthermore assume that the temporal and the spatial scales of fluid dynamics and of chemical reactions are much larger compared to those of thermodynamics. Thus, we suppose the gas is thermodynamically in equilibrium but chemically in non-equilibrium. In order to simulate the behaviour of such a gas mixture, the Navier-Stokes equations have to be augmented by \((N-1)\) additional transport equations for the \(N\) species [18]-[23]. Hence, we obtain formally the same system like Eq. (2.19), but now with the vectors of the conservative variables \(\vec{W}\), the flux vectors \(\vec{F}_c\) and \(\vec{F}_v\), as well as with the source term \(\vec{Q}\) extended by \((N-1)\) species equations. Recalling the expressions (2.20) to (2.25), the vector of the conservative variables reads now

The convective and the viscous flux vectors transform into

where

Finally, the source term now becomes

In the above expressions Eqs. (2.32)-(2.35), \(Y_m\) denotes the mass fraction, \(h_m\) the enthalpy, and \(D_m\) the effective binary diffusivity of species \(m\), respectively. Furthermore, \(\dot{s}_m\) is the rate of change of species \(m\) due to chemical reactions. Note that the total density \(\rho\) of the mixture is equal to the sum of the densities of the species \(\rho Y_m\). Therefore, since the total density is regarded as an independent quantity, there are only \((N-1)\) independent densities \(\rho Y_m\) left. The remaining mass fraction \(Y_N\) is obtained from

In order to find an expression for the pressure \(p\), we first assume that the individual species behave like ideal gases, i.e.,

with \(R_u\) denoting the universal gas constant and \(W_m\) being the molecular weight, respectively. Together with Dalton's law,

we can write

It is important to notice that because the gas is in thermodynamical equilibrium, all species possess the same temperature \(T\). The temperature has to be calculated iteratively from the expression [21], [24]

The internal energy of the gas mixture \(e\) is obtained from Eq. (2.6). The quantities \(h^{0}_{f,m}\), \(c_{p,m}\), and \(T_{ref}\) in Eq. (2.40) denote the heat of formation, the specific heat at constant pressure, and the reference temperature of the \(m\)-th species, respectively. Values of the above quantities as well as of the thermal conductivity \(k\) and of the dynamic viscosity \(\mu\) of the species are determined from curve fits [19], [21], [23].

The last part, which remains to be modelled, is the chemical source term \(\dot{s}_m\) in Eq. (2.35). The rate equations for a set of \(N_R\) elementary reactions involving \(N\) species can be written in the general form

In the above Eq. (2.41), \(\nu'_{lm}\) and \(\nu''_{lm}\) are the stoichiometric coefficients for species \(m\) in the \(l\)-th forward and backward reaction, respectively. Furthermore, \(C_m\) stands for the molar concentration of species \(m\) (\(C_m = \rho Y_m/W_m\)), and finally \(K_{fl}\) and \(K_{bl}\), respectively, denote the forward and the backward reaction rate constants for the \(l\)-th reaction step. They are given by the empirical Arrhenius formulae

where \(A_f\) and \(A_b\) are the Arrhenius coefficients, \(E_f\) and \(E_b\) represent the activation energies, and \(B_f\) as well as \(B_b\) are constants, respectively. The rate of change of molar concentration of species \(m\) by the \(l\)-th reaction is given by

Hence, together with Eq. (2.43) we can calculate the total rate of change of species \(m\) from

More details can be found in the references cited above. A detailed overview of the equations governing a chemically reacting flow, together with the Jacobian matrices of the fluxes and their eigenvalues, can also be found in [24].

Another practical example of real gas is the simulation of steam or, which is more demanding, of wet steam in turbomachinery applications [25]-[32]. In the later case, where the steam is mixed with water droplets, so that we speak of multiphase flow, it is either possible to solve an additional set of transport equations, or to trace the water droplets along a number of streamlines. These simulations have very important applications in the design of modern steam turbine cascades. The analysis of flow past turbine blades can for instance help to understand the occurrence of supercritical shocks by condensation and of flow instabilities, responsible for an additional dynamic load on the bladings and resulting in a loss of the efficiency.

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.