Parabolized Navier-Stokes Equations Research Articles

Fully Parabolized Hypersonic Sonic Boom Prediction with Real Gas and Viscous Effects

We present a methodology to predict the aerodynamic near-field and sonic boom signature from slender bodies and waveriders using a fully parabolized approach. We solve the parabolized Navier–Stokes equations, which are integrated via spatial marching in the streamwise direction. We find that unique physics must be accounted for in the hypersonic regime relative to the supersonic, which includes viscous, nonequilibrium, and real gas effects. The near-field aerodynamic pressure is propagated through the atmosphere to the ground via the waveform parameter method. To illustrate the approach, three bodies are analyzed: the Sears–Haack geometry, the HIFiRE-5, and a power-law waverider. Ambient Mach numbers range from 4 through 15. The viscous stress tensor is essential for accurate hypersonic prediction. For example, viscous effects increase near-field and sonic boom overpressure by 15.7 and 8.49%, respectively, for the Sears–Haack geometry. The difference between viscous and inviscid predictions of the near-field is due to the hypersonic boundary layer. The computational cost for predicting the near-field is approximately 6.6% relative to fully nonlinear computational fluid dynamics.

Numerical simulation of compressible gas flow in flat channels in the narrow channel approximation

In this paper, numerical simulation of a compressible gas in plane channels of constant and variable cross-section is performed within the framework of two-dimensional parabolized Navier-Stokes equations. The "narrow channel approximation" model is used for the numerical solution of the uranation system.A number of transformations are described in detail, such as de-dimensioning the equation system, reducing the considered area to a square, as well as thickening the calculated points in large gradients of gas dynamic parameters. Flow conservation conditions are used to determine the pressure gradient.An effective method is given for simultaneously determining the pressure gradient and the longitudinal velocity, then other gas dynamic parameters stable for subsonic and supersonic flows, as well as a method for determining the critical flow rate for solving Laval nozzle problems.The results of methodical calculations are presented, with the aim of verifying the effectiveness of the developed calculation methodology, as well as confirming the reliability of the results obtained by comparing them with data from other authors

Far Field of a Three-Dimensional Laminar Wall Jet

We consider a steady submerged laminar jet of viscous incompressible fluid flowing out of a tube and propagating along a solid plane surface. The numerical solution of Navier–Stokes equations is obtained in the stationary three-dimensional formulation. The hypothesis that at large distances from the tube exit the flowfield is described by the self-similar solution of the parabolized Navier–Stokes equations is confirmed. The asymptotic expansions of the self-similar solution are obtained for small and large values of the coordinate in the jet cross-section. Using the numerical solution the self-similarity exponent is determined. An explicit dependence of the self-similar solution on the Reynolds number and the conditions in the jet source is determined.

Solution of Parapolic Navier-Stokes Equations for Laminar Forced Convective Flow in Entrance Region of a Flat Passage.(Dept.M)

Due to dependence of forced convective heat transfer on the hydrodynamic flow field; any improvement on the analysis of this flow field is of great importance to understand the heat transfer process. In this work the flow field is described by the parabolic Navier-Stokes equations; which are solved by local non-similarity solution-method. According to this method; with suitable definition of the problem variables, a set of ordinary differential equations are produced. This set is solved, numerically, by Runge-Kutta method accompanied with shooting method of boundary value problems. The values of local Nusselt number and local coefficient of friction are calculated for fluids of prandtl number of one. Some passages of Reynolds number (Re1) of 100, 200 & 300 are studied here. Two formulate of Nusselt number for different passage heights are proposed.

Solution of Parabolic Navier-Stokes Equations for The Entrance Region-Flow between Two Parallel Plates.(Dept.M)

The development of the flow field in the entrance region is theoretically examined, for the case of flow between two parallel flat plates. The flow field is described by the parabolic Navier-Stokes equations. Because of the nature of these equations; it is convenient to solve them by the application of the local similarity solution method. According to this method, dimensionless form of momentum equations are transferred to ordinary differential equations. These two modified equations are solved numerically by Runge-Kutta method of ordinary differential equations accompanied with shooting method of boundary value problems. The values of coefficient of friction are calculated at different positions along the passage. Also the velocity profile at any position, according to this approach, can be obtained. Three passages, in this work, are studied; with Reynolds number ( based on the half of the height of the passage) of 100, 200 and 300.

Laminar and Turbulent Characteristics of the Acoustic/Fluid Dynamics Interactions in a Slender Simulated Solid Rocket Motor Chamber

In this paper, analytical, computational, and experimental studies are integrated to examine unsteady acoustic/vorticity transport phenomena in a solid rocket motor chamber with end-wall disturbance and side-wall injection. Acoustic-fluid dynamic interactions across the chamber may generate intense unsteady vorticity with associated shear stresses. These stresses may cause scouring and, in turn, enhance the heat rate and erosional burning of solid propellant in a real rocket chamber. In this modelling, the unsteady propellant gasification is mimicked by steady-state flow disturbed by end-wall oscillations. The analytical approach is formulated using an asymptotic technique to reduce the full governing equations. The equations that arise from the analysis possess wave properties are solved in an initial-boundary value sense. The numerical study is performed by solving the parabolized Navier–Stokes equations for the DNS simulation and unsteady Reynolds-averaged Navier–Stokes equations along with the energy equation using the control volume approach based on a staggered grid system with the turbulence modelling. The v2-f turbulence model has been implemented. The results show that an unexpectedly large amplitude of unsteady vorticity is generated at the injection side-wall of the chamber and is then penetrated downstream by the bulk motion of the internal flow. These stresses may cause a scouring effect and large transient heat transfer on the combustion surface. A comparison between the analytical, computational, and experimental results is performed.

Theoretical model of swirling thick film flow inside converging nozzles of various geometries

Pressure-swirl jets are of paramount importance in such applications as liquid rockets, gas turbines, and internal combustion engines. Although the external spray has been extensively studied, the internal nozzle flow, which determines the external spray characteristics outside the nozzle, is relatively underexplored. Experimental studies of the internal nozzle flow are available, whereas theoretical approaches are less common, especially for strong swirl flows in which the boundary layer effect at the nozzle walls is significant. Herein, we explore a strongly swirling film flow, where viscous effects become dominant in the entire film and the boundary-layer approximation fails. The parabolized Navier-Stokes equations are solved by using the integral von Karman-Pohlhausen method, and the viscous liquid film flow over the entire nozzle wall is predicted including the central free surface configuration, the film characteristics such as its thickness, velocity components, and the spray cone angle, at the nozzle exit. The theoretical predictions are compared to the experimental data, and the dependence of the spray-cone angle on mass flow rate is experimentally validated. The theoretical parametric studies include the effect of the internal nozzle geometry, initial film thickness, the mass flow rate, and the effect of the swirling strength.

Application of the Implicit MacCormack Scheme for Computation of Supersonic Turbulent Jets Using the Parabolized Navier—Stokes Equations

An effective unconditionally stable numerical method for solving equations describing the flow of supersonic turbulent jets is developed. The method does not require the inversion of block tridiagonal systems of algebraic equations. The results obtained according to this method are in fairly close agreement with the calculations results on the basis of other methods and experimental data.

Stability analysis of MacCormack rapid solver method for evolutionary Stokes–Darcy problem

A numerical method based on the explicit–implicit scheme of MacCormack finite difference scheme was applied to the solution of Parabolized Navier–Stokes equations. Partitioned methods, which solve the coupled problem by successively solving the sub-physics problems, have recently been studied for the full evolutionary groundwater–surface water flow with convergence established over both bounded and long time intervals. A MacCormack rapid solver method based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model. Under a time step limitation of the form C1h≤Δt≤C2h (where h, Δt are mesh size and time step, respectively, and Cj, j∈{1,2}, are two physical parameters) we prove both uniform and asymptotic stabilities over long time intervals. Some numerical experiments are presented and discussed.

An effective numerical method for calculating unseparated flows in building aerodynamics

The problem of the mixing of heated gases is of interest in connection with the development of pollution-free technologies for natural-fuel combustion in modern heat and electricity generating plants. The operating principle of the tall structures designed for this purpose, which combine a smokestack and a cooling tower, is as follows. At the base of the stack flue gas, from which the sulfur has been removed, is fed into a flow of air heated in a heat exchanger. As it moves through the stack, the gas mixes with the hot air and is carried into the atmosphere by the natural draft. The design must satisfy certain requirements. The temperature of the flue gas must not fall below a certain limit at which condensation that leads to corrosion develops. The gas outlet velocity must be higher than 4 m/s to prevent downdraft. The concentrations of pollutants released into the atmosphere must be within the permissible limits. One promising means of enhancing the efficiency of such structures is to swirl the flue gas ahead of the stack inlet. Flow swirling considerably intensifies the heat and mass transfer processes, improves the mixing of the hot gases, reduces the pollutant concentrations at the stack outlet, and prevents flow separation on the walls. The general formulation of the problem of the mixing of two nonisothermal turbulent flows is based on the complete Reynolds equations. This system closed by a turbulence model (algebraic or differential) is fairly complicated and its solution is laborious. A simplified model is based on the parabolized Navier-Stokes equations, which restricts the area of applicability to unseparated flows. However, in view of the mechanical nature of the problem considered, unseparated flows are of most interest. In this paper, an effective method of solving boundary-layer equations is discussed, in which the initial system reduces to ordinary differential equations written on the streamlines.

Numerical analysis of combustion of a hydrogen–air mixture in an advanced ramjet combustor model during activation of O2 molecules by resonant laser radiation

This paper presents a numerical study of the combustion of a hydrogen–air mixture in a model ramjet combustor with separate hydrogen and air supply during activation of O2 molecules by resonant laser radiation at a wavelength of 762.3 nm and 193.3 nm. The calculation is made using the parabolized Navier–Stokes equations taking into account chemical reactions, laser irradiation, and the nonuniformity of air parameters at the combustor inlet due to the complex gas-dynamic structure of the flow in the air intake. It is shown that the combustion completeness at the combustor outlet can be increased by a factor of 2.8 by redistributing the hydrogen supply through the system of fuel tank pylons. Further increase in the combustion completeness can be obtained by exposure of a narrow flow region to resonant laser radiation, more effectively at a wavelength of 193.3 nm. The combination of laser exposure with hydrogen supply redistribution increases the combustion efficiency by a factor of more than 4.7 compared to the base case. In this case, this provides a 95% increase the longitudinal force component in the portion of the internal engine duct that provides a positive contribution to the thrust. Estimation of the energy efficiency of using laser radiation shows that the laser energy input required to achieve this effect is 40–80 times (depending on the fuel supply method) less than the increase in the chemical energy (compared to the case of no laser exposure) released due to fuel combustion.

A stabilized Crank-Nicolson mixed finite element method for non-stationary parabolized Navier-Stokes equations

In this study, a time semi-discrete Crank-Nicolson (CN) formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite element (SCNMFE) formulation based on two local Gauss integrals and parameterfree with the second-order time accuracy is established directly from the time semi-discrete CN formulation. Thus, it could avoid the discussion for semi-discrete SCNMFE formulation with respect to spatial variables and its theoretical analysis becomes very simple. Finaly, the error estimates of SCNMFE solutions are provided.

A combination of parabolized Navier–Stokes equations and level-set method for stratified two-phase internal flow

A novel methodology is proposed for the numerical computation of pressure-driven gravity-stratified flows along channels comprising two immiscible phases. The parabolized Navier–Stokes equations are combined with the level set approach, resulting into a downstream-marching problem in which the solution is computed at each cross-section based on upstream information only. A main difficulty in the implementation of the approach for internal flows is the conservation of the mass flow rates, which is addressed by extending to two-phase flows the method proposed by Patankar and Spalding (1972) and Raythby and Schneider (1979), and by adding an explicit forcing term in the equation for the advection of the level function. The combination of high-order finite differences and sparse storage and algebra used here allows a fully-coupled integration of the parabolized equations, as opposed to the more classical segregated approaches. This enables a very efficient calculation of the complete downstream-developing flow field.

Study of effect of a smooth hump on hypersonic boundary layer instability

Effect of a two-dimensional smooth hump on linear instability of hypersonic boundary layer is studied by using parabolized stability equations. Linear evolution of mode S over a hump is analyzed for Mach 4.5 and 5.92 flat plate and Mach 7.1 sharp cone boundary layers. Mean flow for stability analysis is obtained by solving the parabolized Navier–Stokes equations. Hump with height smaller than local boundary layer thickness is considered. The case of flat plate and sharp cone without the hump are also studied to provide comparable data. For flat plate boundary layers, destabilization and stabilization effect is confirmed for hump located at upstream and downstream of synchronization point, respectively. Results of parametric studies to examine the effect of hump height, location, etc., are also given. For sharp cone boundary layer, stabilization influence of hump is also identified for a specific range of frequency. Stabilization influence of hump on convective instability of mode S is found to be a possible cause of previous experimental observations of delaying transition in hypersonic boundary layers.

A stabilized Crank-Nicolson mixed finite volume element formulation for the non-stationary parabolized Navier-Stokes equations

A time semi-discrete Crank-Nicolson (CN) formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite volume element (SCNMFVE) formulation based on two local Gaussian integrals and parameter-free with the second-order time accuracy is established directly from the time semi-discrete CN formulation so that it could avoid the discussion for semi-discrete SCNMFVE formulation with respect to spatial variables and its theoretical analysis becomes very simple. Finally, the error estimates of SCNMFVE solutions are provided.

A POD-BASED REDUCED-ORDER STABILIZED CRANK–NICOLSON MFE FORMULATION FOR THE NON-STATIONARY PARABOLIZED NAVIER–STOKES EQUATIONS

We firstly employ a proper orthogonal decomposition (POD) method, Crank–Nicolson (CN) technique, and two local Gaussian integrals to establish a PODbased reduced-order stabilized CN mixed finite element (SCNMFE) formulation with very few degrees of freedom for non-stationary parabolized Navier–Stokes equations. Then, the error estimates of the reduced-order SCNMFE solutions, which are acted as a suggestion for choosing number of POD basis and a criterion for updating POD basis, and the algorithm implementation for the POD-based reduced-order SCNMFE formulation are provided, respectively. Finally, some numerical experiments are presented to illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order SCNMFE formulation is feasible and efficient for finding numerical solutions of the non-stationary parabolized Navier–Stokes equations.

Effect of bulk viscosity in supersonic flow past spacecraft

In this paper, we consider the effect of bulk viscosity in various hydrodynamic problems. We numerically study this effect on the front structure of the one-dimensional stationary shock wave and on the flow past blunt body. We estimate the effect of the bulk viscosity coefficient (BVC) on the heat transfer and drag of a sphere in a supersonic flow, apparently for the first time, by the numerical solution of parabolized Navier–Stokes equations. The solution is obtained by an original fast convergent method of global iterations of the longitudinal pressure gradient. The directions of further investigations of bulk viscosity are suggested.

Three-Dimensional Solutions of Trailing-Vortex Flows Using Parabolized Equations

Flows of practical significance exist, like systems of trailing vortices, which are inhomogeneous in two directions and weakly dependent along the third spatial direction. Exploiting these characteristics, an integration of the Navier–Stokes equations using the parabolic Navier–Stokes concept is proposed for the recovery of steady solutions that might be used subsequently in the scope of primary instability analyses. The parabolic Navier–Stokes equations are first formulated in a cylindrical coordinate frame and used to calculate the solution of an isolated, axisymmetric nonparallel (axially developing) vortex. Then, the fully three-dimensional flow corresponding to a counter-rotating pair of nonparallel vortices is obtained by parabolic Navier–Stokes formulated in Cartesian coordinates. Stable high-order finite-difference-based numerical schemes are used for the spatial discretization to exploit the benefits of using a sparse direct solver for the inversion of the large matrices resulting from the discretization of the partial-derivative-based parabolic Navier–Stokes equations. Solutions recovered converge to the analytically obtained evolution of an isolated vortex in the limit of large distance between vortices and depart from this idealized situation when the distance becomes of the same order as the radii of the vortices.

An approximate Riemann solver for real gas parabolized Navier–Stokes equations

Under specific assumptions, parabolized Navier–Stokes equations are a suitable mean to study channel flows. A special case is that of high pressure flow of real gases in cooling channels where large crosswise gradients of thermophysical properties occur. To solve the parabolized Navier–Stokes equations by a space marching approach, the hyperbolicity of the system of governing equations is obtained, even for very low Mach number flow, by recasting equations such that the streamwise pressure gradient is considered as a source term. For this system of equations an approximate Roe’s Riemann solver is developed as the core of a Godunov type finite volume algorithm. The properties of the approximated Riemann solver, which is a modification of Roe’s Riemann solver for the parabolized Navier–Stokes equations, are presented and discussed with emphasis given to its original features introduced to handle fluids governed by a generic real gas EoS. Sample solutions are obtained for low Mach number high compressible flows of transcritical methane, heated in straight long channels, to prove the solver ability to describe flows dominated by complex thermodynamic phenomena.

Reduced order modeling based on POD of a parabolized Navier–Stokes equations model II: Trust region POD 4D VAR data assimilation

A reduced order model based on Proper Orthogonal Decomposition (POD) 4D VAR (Four-dimensional Variational) data assimilation for the parabolized Navier–Stokes (PNS) equations is derived. Various approaches of POD implementation of the reduced order inverse problem are studied and compared including an ad-hoc POD adaptivity along with a trust region POD adaptivity. The numerical results obtained show that the trust region POD 4D VAR provides the best results amongst all the POD adaptive methods tested in all error metrics for the reduced order inverse problem of the PNS equations.