Velocity coupling

Because any such behavior causes the motor to fail in its mission objective, these peculiar operational effects have received considerable research attention. The results of these research studies have shown that these various forms of instability result from a coupling between the transient combustion characteristics of the propellant and the transient ballistics of the combustion chamber. These instabilities are termed pressure-coupled, velocity-coupled, and bulk-coupled, and will be described below. 

Church s equations, 176-177 Combustion instability, 52-57 bulk-coupled, 56-57 pressure-coupled, 52-55 velocity-coupled, 55-56 steady-state, 29-51 prediction, 30 pressure plateaus, 34 propellants, 31-50 termination, 57-64 depressurization, 58-62 fluid-injection, 63-64 L, 62-63... 

For the computation of compressible flow, the pressure-velocity coupling schemes previously described can be extended to pressure-velocity-density coupling schemes. Again, a solution of the linearized, compressible momentum equation obtained with the pressure and density values taken from a previous solver iteration in general does not satisfy the mass balance equation. In order to balance the mass fluxes into each volume element, a pressure, density and velocity correction on top of the old values is computed. Typically, the detailed algorithms for performing this task rely on the same approximations such as the SIMPLE or SIMPLEC schemes outlined in the previous paragraph. 

Fig. 13.21 shows another example of oscillatory burning of an RDX-AP composite propellant containing 0.40% A1 particles. The combustion pressure chosen for the burning was 4.5 MPa. The DC component trace indicates that the onset of the instability is 0.31 s after ignition, and that the instability lasts for 0.67 s. The pressure instability then suddenly ceases and the pressure returns to the designed pressure of 4.5 MPa. Close examination of the anomalous bandpass-filtered pressure traces reveals that the excited frequencies in the circular port are between 10 kHz and 30 kHz. The AC components below 10 kHz and above 30 kHz are not excited, as shown in Fig. 13.21. The frequency spectrum of the observed combustion instability is shown in Fig. 13.22. Here, the calculated frequency of the standing waves in the rocket motor is shown as a function of the inner diameter of the port and frequency. The sonic speed is assumed to be 1000 m s and I = 0.25 m. The most excited frequency is 25 kHz, followed by 18 kHz and 32 kHz. When the observed frequencies are compared with the calculated acoustic frequencies shown in Fig. 13.23, the dominant frequency is seen to be that of the first radial mode, with possible inclusion of the second and third tangential modes. The increased DC pressure between 0.31 s and 0.67 s is considered to be caused by a velocity-coupled oscillatory combustion. Such a velocity-coupled oscillation tends to induce erosive burning along the port surface. The maximum amplitude of the AC component pressure is 3.67 MPa between 20 kHz and 30 kHz. - ...

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

A number of practical situations involve convection heat transfer which is neither forced nor free in nature. The circumstances arise when a fluid is forced over a heated surface at a rather low velocity. Coupled with the forced-flow velocity is a convective velocity which is generated by the buoyancy forces resulting from a reduction in fluid density near the heated surface. 

Example 9.9 Chemical reaction velocity coupled to mass flow For the vanishing cross coefficients Liq and Lqi in Eqs. (9.116)—(9.118), the heat flow and the reaction velocity become... 

Example 9.10 Chemical reaction velocity coupled to heat flow In this case, LSl and LlS vanish. Still, heat and mass flows are coupled. The new phenomenological equations are... 

Nonlinear phenomena, usually associated with high amplitudes of the acoustic field, can introduce many interesting effects into acoustic instability [76]. Here we shall discuss only three topics involving nonlinearity the response of the combustion zone to transverse velocity oscillations (conventionally termed velocity coupling), changes in the mean burning rate of the propellant in the presence of an acoustic field, and instabilities that involve the propagation of steep-fronted waves (identified in the introduction as shock instabilities). 

Jang, D.S., Jetli, R. and Acharya, S. (1986), Comparison of the PISO, SIMPLER and SIMPLEC algorithms for the treatment of the pressure velocity coupling in steady flow problems. Numerical Heat Transfer, 19, 209-228. 

In this approach, the finite volume methods discussed in the previous chapter can be applied to simulate the continuous fluid (in a Eulerian framework). Various algorithms for treating pressure-velocity coupling, and the discussion on other numerical issues like discretization schemes are applicable. The usual interpolation practices (discussed in the previous chapter) can be used. When solving equations of motion for a continuous fluid in the presence of the dispersed phase, the major differences will be (1) consideration of phase volume fraction in calculation of convective and diffusive terms, and (2) calculation of additional source terms due to the presence of dispersed phase particles. For the calculation of phase volume fraction and additional source terms due to dispersed phase particles, it is necessary to calculate trajectories of the dispersed phase particles, in addition to solving the equations of motion of the continuous phase. 

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