Exit Boundary Conditions Research Articles

Prediction of Turbulent Flow and Heat Transfer in a Radially Rotating Square Duct

The present study deals with the numerical prediction of turbulent flow and heat transfer in a rotating duct of square cross section. The axis of rotation is normal to the axis of the duct, and the flow is radially outward. The duct is smooth, of finite length, and the walls are isothermal at a temperature greater than the temperature of the incoming fluid. Both the Coriolis and the centrifugal-buoyancy effects are considered; the problem is three dimensional and fully elliptic. The predicted flow field is found to be quite complex, consisting of secondary cross-stream flows due to the Coriolis effects. Centrifugal buoyancy increases the radial velocity of the cooler fluid near the trailing face and decreases the radial velocity of the warmer fluid near the leading face; indeed, when the buoyancy effects are sufficiently strong, reverse radial flow may occur over the leading face. Rotation is found to increase the heat transfer over the trailing face, while, over the leading face, the heat transfer decreases near the inlet but increases further downstream. This finding agrees with the experimental observations. The quantitative agreement with the data is also satisfactory. The predictions are found to be quite sensitive to the inlet conditions, in particular, to the presence of rotational effects in the incoming stream. The effect of exit boundary conditions is examined by comparing the predictions for a single passage with those for a double-leg passage.

Numerical analysis of flow through oscillating cascade sections

The design of turbomachinery blades requires the prevention of flutter for all operating conditions. However, flow field predictions used for aeroelastic analysis are not well understood for all flow regimes. The present research focuses on numerical solutions of the Euler and Navier-Stokes equations using an ADI procedure to model two-dimensional, transonic flow through oscillating cascades. The model prescribes harmonic pitching motions for the blade sections for both zero and nonzero interblade phase angles. The code introduces the use of a deforming grid technique for convenient specification of the perioidic boundary conditions. Approximate nonreflecting boundary conditons were coded for the inlet and exit boundary conditions. Sample unsteady solutions were performed for an oscillating cascade and compared to experimental data. Also, test cases were run for a flat plate cascade to compare with the unsteady, small-perturbation, subsonic analysis. The predictions for oscillating cascades with nonzero interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. The zero degree interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. Studies on reflecting versus nonreflecting inlet and exit boundary conditions show that the treatment of the boundary can have a significant effect on the first harmonic, unsteady pressure distribution for certain flow conditions.

Transonic Cascade Flow Solved by the Combined Shock-Capturing and Shock-Fitting Method

A method for solving steady transonic flows has been developed that is a combination of the shock-capturing and shock-fitting methods and takes full advantage of each of these two approaches. In the calculation procedure, by use of the artificial compressibility and the approximate factorization scheme for transonic flowfields (AF2), a full-potential equation is used first to calculate the whole blade-to-blade flowfield and smeared shock waves are captured automatically as regions of steep gradients in flow variables. Subsequently, the definite position of shocks is fitted from these regions based on the shock relations. The Kutta and exit boundary conditions are modified to allow for the effect of entropy variation, and the nonisentropic potential formulation is utilized to solve the flow after the shocks. The numerical experiments show that the present method can automatically give clear and sharp shock discontinuities and reasonable distributions of gas variables. The calculation time is only about twice that needed by a standard potential solution. Hence, this method is effective as well as economical

Reacting solid particles in one-dimensional nozzle flow

A fully conservative explicit scheme for a solid-particle-laden gas flow in a convergent-divergent nozzle is studied numerically as a direct time-dependent problem for a one-dimensional flow. The initial conditions at t = 0 are taken from the similarity solution. The boundary conditions at the nozzle inlet are determined by taking an arbitrary large cross-section at the first grid point before the nozzle inlet and are held constant with time. The exit boundary conditions are obtained at the grid point next to the exit plane from the values extrapolated at the exit plane. The particle burning rate is assumed to be a function of pressure. A time-dependent predictor-corrector formulation with a fourth-order damping term is used. Convergence of the result is assumed when the throat Mach number change is <0.0001, generally in ⩽ 400 time steps.

Solution of the advection‐dispersion equation with free exit boundary

AbstractI investigated the exit boundary condition for the advection‐dispersion equation and found that in numerical solutions of this equation, using Galekin finite elements, a free exit boundary condition requiring no a priori information is possible, provided the advective component in the numerical equations is of sufficient magnitude relative to the dispersive component. Since the relationship between these two components is controlled by the spatial discretization through the grid Peclet number, the free exit boundary condition can in fact be applied whenever there is a non‐zero advective component. The numerical solution in a finite domain with free exit boundary, using a correctly proportioned spatial discrezation, behaves like an infinite‐domain solution.

Application of the Convection‐Dispersion Model to Solute Transport in Finite Soil Columns

AbstractThe one‐dimensional convection‐dispersion model is investigated with respect to the proper boundary conditions to impose in applications to tracer movement in freely draining soil columns. It is shown that the model can be interpreted probabilistically and that this interpretation leads to physical restrictions on solute molecule transport through the inlet and exit boundaries of a soil column. These constraints, in turn, show that the correct exit boundary condition for a freely draining soil column is zero resident concentration or, equivalently, zero gradient in the flux concentration. This boundary constraint is combined with conditions appropriate to either an open or a closed inlet boundary to derive a variety of mass‐conserving solutions (some apparently new) to the convection‐dispersion model for both resident and flux concentrations. In particular, a new solution is presented for the resident concentration under a closed inlet boundary. The corresponding flux concentration solution for a constant solute flux at the inlet is shown to be the same as a solution first obtained by R.W. Cleary and D.D. Adrian (1973).

Identification of longitudinal acoustic modes associated with pressure oscillations in ramjets

A one-dimensiona l, isentropic analytical method is presented for calculating the longitudinal acoustic modes in an idealized dump-type ramjet engine. The analysis accounts for the complex acoustic admittances of the choked exit nozzle and of the normal shock in the supercritical inlet. The method assumes that all mean flow properties such as inlet and combustor temperatures, densities, and average Mach numbers are known. The predicted longitudinal modes are compared with data acquired from three different types of experiments. The first experiment concerned a wind tunnel model of a missile which utilized a ramjet with two side-dump inlets. The wind tunnel tests were primarily intended to determine the behavior of the supersonic inlets in the presence of downstream pressure oscillations. A mechanical device was used to excite periodic pressure perturbations in the simulated combustor exit. The second experiment involved connected pipe combustion tests of a side dump combustor. These tests included systematic changes in combustor length and in the entrance and exit boundary conditions. The third experiment utilized connected pipe tests of a coaxial dump combustor. The geometric length of the combustor was varied in order to examine the effects on combustion instability.

Influence of Exit Impedance on Finite Difference Solutions of Transient Acoustic Mode Propagation in Ducts

The time-dependent governing acoustic-difference equations and boundary conditions are developed and solved for sound propagation in an axisymmetric (cylindrical) hard-wall duct without flow and with spinning acoustic modes. The analysis begins with a harmonic sound source radiating into a quiescent duct. This explicit iteration method then calculates stepwise in real time to obtain the steady solutions of the acoustic field. The transient method did not converge to the steady-state solution for cutoff acoustic duct modes. This has implications as to its use in a variable-area duct, where modes may become cutoff in the smal-area portion of the duct. For single cutoff mode propagation the steady-state impedance boundary condition produced acoustic reflections during the initial transient that caused finite instabilities in the numerical calculations. The stability problem is resolved by reformulating the exit boundary condition. Example calculations show good agreement with exact analytical and numerical results for forcing frequencies above, below, and nearly at the cutoff frequency.

A crossflow model of dispersion in packed bed reactors

AbstractA three‐parameter model of axial and radial dispersion in a packed tubular reactor is proposed as an alternative to the Fickian dispersion model. Though it is formulated in terms of differential equations, it requires no exit boundary condition, so it is mathematically more convenient than the Fickian model. Examples of its application to chemical reactors are given.

Stability of radiation-heated flow

The stability of subsonic flow which is radiation-heated and subsequently discharged through a supersonic nozzle is analyzed. A solution is found for steady, one-dimensional flow and the equations for small harmonic disturbances are derived. The transfer function relating back pressure to flow momentum is found by integration considering the sonic and exit boundary conditions. A characteristic equation using this transfer function is developed and used to demonstrate stability.

Axisymmetric Mixing Layer: Influence of the Initial and Boundary Conditions

The mixing layer in the near field of a 12.7-cm-diam circular air jet has been experimentally investigated at a jet speed of 30 m/s. Data have been obtained with a computer-trave rsed hot wire for laminar as well as equilibrium turbulent exit boundary layers, both without and with a large end plate. The integral measures of the mixing layer and their streamwise evolutions show essentially no dependence on the end plane geometry (i.e., exit boundary condition) except for a small A range (x/0e — 500) when initially laminar, but show strong dependence on the initial condition (i.e., laminar vs turbulent). The mean velocity and turbulence intensity profiles and streamwise spectral evolutions indicate that the shear layers have achieved self-preservation. Initially turbulent layers show two stages of linear growth; compared to the initially laminar layers, the growth rate is lower in the first stage but higher in the second stage. The turbulent field achieves self-similarity much later than the mean field when the initial state is turbulent, but essentially together when the initial state is laminar. The virtual origin is upstream of the lip for the initially laminar layers, but downstream for initially turbulent layers. Suprisingly, the present data are in both qualitative and quantitative agreement with plane mixing layer results. The Rex range (up to 10 6) in this study is apparently insufficient to resolve the question of the asymptotic effect of the initial condition of a mixing layer.

Dynamic Blade Row Compression Component Model for Stability Studies

This paper describes a generalized dynamic model which has been developed for use in compression component aerodynamic stability studies. The model is a one-dimensional, pitch-line, blade row, lumped volume system. Arbitrary placement of blade free volumes upstream, within, and downstream of the compression component as well as the removal of bleed flow from the exit of any rotor or stator are model options. The model has been applied to a two-stage fan and an eight-stage compressor. The clean inlet pressure ratio/flow maps and the surge line have been reproduced, thereby validating the capability of the dynamic model to reproduce the steady-flow characteristics of the compression component. A method for determining the onset of an aerodynamic instability which is associated with surge is described. Sinusoidally time-varying inlet and exit boundary conditions have been applied to the eight stage compressor as examples of the manner in which this model may be used for stability studies.

Numerical Representation of Inlet and Exit Boundary Conditions in Transient Cascade Flow

An important building block in the development of finite difference time dependent techniques for mixed supersonic-subsonic flow calculation is the treatment of the inlet and exit boundary conditions. For cascade computations, it is convenient to locate the planes at a finite distance from the blade and hence conditions along these planes must be allowed to vary according to the overall cascade performance. It is shown in this paper that as long as the meridional velocity is subsonic, three flow variables must be specified at the inlet and one variable at the exit. The remaining flow variables must be computed as part of the solution. The paper contains a method derived from the theory of characteristics for the computation of the remaining variables under the assumption that the flow is axisymmetric at the inlet and exit planes. The response of typical turbine and compressor cascades to back pressure variations as computed by this technique is also presented.

Shock stability in magnetogasdynamic channel flows

The question of shock stability in a perfect-gas channel flow was examined in [1] in the onedimensional approximation under various assumptions: the disturbances are not reflected from the channel exit section, weak shock, etc. The results were found to coincide for two specific forms of the boundary conditions at the channel exit, from which it was concluded that the shock was not sensitive to the exit boundary condition. In [2] the question of shock stability was studied numerically in relation to a conducting-gas flow in a flat channel of constant cross section in the presence of a magnetic field (zero electric field intensity). It was established that the shock stability is significantly affected by the form of the conductivity law. A condition for the limiting regime between the stable and unstable regions was also given for flow with a shock wave. It was assumed that the pressure in the channel exit section is given. In this paper the effect of the exit boundary condition on shock stability in gasdynamic and magnetogasdynamic flows is demonstrated for small magnetic Reynolds numbers. Stability criteria are obtained for shocks near the channel exit for a specific exit condition. The influence of electromagnetic effects (conductivity law, electric load factor) on shock stability is investigated.