Stagnation flow compressible

In the steady stagnation-flow formulation the thermodymanic pressure may be assumed to be constant and treated as a specified parameter. The small pressure variations in the axial direction, which may be determined from the axial momentum equaiton, become decoupled from the system of governing equations (Section 6.2). The small radial pressure variations associated with the pressure-curvature eigenvalue A are also presumed to be negligible. While this formulation works very well for the steady-state problem, it can lead to significant numerical difficulties in the transient case. A compressible formulation that retains the pressure as a dependent variable (not a fixed parameter) relieves the problem [323],... 

Deriving the compressible, transient form of the stagnation-flow equations follows a procudeure that is largely analogous to the steady-state or the constant-pressure situation. Beginning with the full axisymmetric conservation equations, it is conjectured that the solutions are functions of time t and the axial coordinate z in the following form axial velocity u = u(t, z), scaled radial velocity V(t, z) = v/r, temperature T = T(t, z), and mass fractions y = Yk(t,z). Boundary condition, which are applied at extremeties of the z domain, are radially independent. After some manipulation of the momentum equations, it can be shown that... 

Fig. 17.14 Finite-volume, staggered-grid, spatial-difference stencil for the transient compressible stagnation-flow equations. Grid points, which are at control-volume centers, are used to represent all dependent variables except axial velocity, which is represented at the control-volume faces. The grid indexes are shown on the left and the face indexes on the right. The right-facing protuberance on the stencils indicates where the time derivative is evaluated. For the pressure-eigenvalue equation there is no time derivative.

With the brief discussion of index, it is now possible to identify and compare some aspects of the high-index behavior of the constant-pressure and the compressible stagnation-flow equations. To understand the structure of the DAE system, it is first necessary to identify all variables that are not time differentiated (i.e., the x vector). In the constant-pressure formulation, neither the axial velocity u nor the pressure curvature A has time derivatives. By introducing the axial momentum equation, the compressible formulation introduces du/dt. To be of value in reducing the index, however, the momentum equation must be coupled to the other equations. The coupling is accomplished through pressure, which is included as a dependent variable. The variable A is not time differentiated in either formulation. 

It may be shown that the DAE system corresponding to the discrete form of the compressible stagnation-flow equations is of the so-called Hessenberg-index-two structure [46], which is represented by Eq. 17.29. The constraints g do not depend on x, and the matrix... 

L.L. Raja, R.J. Kee, and L.R. Petzold. Simulation of the Transient, Compressible, Gas-Dynamic, Behavior of Catalytic-Combustion Ignition in Stagnation Flows. Proc. Combust. Inst., 27 2249-2257,1998. 

Raja LL, Kee RJ, Petzold LR Simulation of the transient, compressible, gas-dynamic behavior of catalytic combustion ignition in stagnation flows, Proc Combust Inst 27 2249—2257, 1998. 

Two-dimensional compressible momentum and energy equations were solved by Asako and Toriyama (2005) to obtain the heat transfer characteristics of gaseous flows in parallel-plate micro-channels. The problem is modeled as a parallel-plate channel, as shown in Fig. 4.19, with a chamber at the stagnation temperature Tstg and the stagnation pressure T stg attached to its upstream section. The flow is assumed to be steady, two-dimensional, and laminar. The fluid is assumed to be an ideal gas. The computations were performed to obtain the adiabatic wall temperature and also to obtain the total temperature of channels with the isothermal walls. The governing equations can be expressed as... 

Probably the most characteristic causative factor in Los Angeles smog is the lid created by the stagnation of warm air subsiding on the cool marine layer. Late in the summer air flows from an anticyclone over the Pacific Ocean and compresses as it loses altitude in its outward course. Onshore movement of marine air under this outflow maintains a... 

However, if the flow in the leak location is subsonic, the hole could be the source of the production of a compression or expansion sonic wave and the effect on the flow field. In this situation, the pipe and leakage spot act as a T junction and the mass outflow of the leak location depends on the ambient pressure and a coefficient known as the empirical discharge coefficient in addition to the stagnation pressure and the orifice area. 

In turbulent compressible flows the total (stagnation) enthalpy, defined as feg = + uj / 2, is offen used as depen-denf variable. Then fhe energy conservafion equafion is generally rewritten as ... 

For a compressible flow, this is the thermodynamic state that would exist if a flow were brought to rest isentropically. In practice, this would correspond to the thermodynamic state of a veiy low-speed flow entering the nozzle inlet from an upstream combustion chamber or a pressurized reservoir. For this reason, stagnation cOTiditions are also sometimes referred to as chamber or reservoir conditions. 

As shown in Figure 22.17, the flow from a tank containing compressible fluid at stagnation pressure Pq to the backpressure P. As the Pj is decreased, the flow rate increases until Pj, reaches the critical value P and then any further decrease in the Pj will not increase the... 

Compressible ID stagnation-point flow analysis forms the basis of the equation system presented below. It was found that the prediction of the effect of internal mass transfer limitations in the catalytic washcoat of the SFR configuration is crucial to derive microkinetic data from SFR experiments (Karadeniz, 2014 Karadeniz et al., 2013) our model is extended to include the diffusion limitations due to a porous layer. It should be noted that the CFiEMKIN code has no abihty to account for internal mass transport in the catalytic coating. 

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