Comparison Between Navier-Stokes and Thin-Layer Computations for Separated Supersonic Flow

An immersed boundary formulation incorporating a two-layer wall model approach for RANS simulations with complex geometry

Up to this point, we have emphasized the second-order centered-difference approximations to the spatial derivatives in our model equations. We have seen that a centered approximation to a first derivative is nondissipative, i.e. the eigenvalues of the associated circulant matrix (with periodic boundary conditions) are pure imaginary. In processes governed by nonlinear equations, such as the Euler and Navier—Stokes equations, there can be a continual production of high-frequency components of the solution, leading, for example, to the formation of shock waves. In a real physical problem, the production of high frequencies is eventually limited by viscosity. However, when we solve the Euler equations numerically, we have neglected viscous effects. Thus the numerical approximation must contain some inherent dissipation to limit the production of high-frequency modes. Although numerical approximations to the Navier—Stokes equations contain dissipation through the viscous terms, this can be insufficient, especially at high Reynolds numbers, due to the limited grid resolution which is practical. Therefore, unless the relevant length scales are resolved, some form of added numerical dissipation is required in the numerical solution of the Navier—Stokes equations as well. Since the addition of numerical dissipation is tantamount to intentionally introducing nonphysical behavior, it must be carefully controlled such that the error introduced is not excessive. In this chapter, we discuss some different ways of adding numerical dissipation to the spatial derivatives in the linear convection equation and hyperbolic systems of PDE’s.KeywordsWave SpeedHyperbolic SystemCourant NumberNumerical DissipationSymmetric ComponentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A boundary element method for analysing non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers has been developed. Employing this method we analyze two kinds of non-linear phenomena on gas-liquid two-phase free surface (l) Solitary wave on free falling thin liquid film. (2) Surface tension drives convective flows in open-boat type crystal-growth techniques in microgravity environment. INTRODUCTION Although some new schemes of discretizing convective terms are developed as QUICK and QUICK ER in finite difference methods[FDM] and finite element methods[FEM], but it is very difficult to avoid numerical diffusion introduced by convective discretizations, especially, at high Reynolds number. In [1], they developed a boundary element iterative scheme(BEIS) to calculate convective terms and compared the advantages of using upwind or central difference schemes and BEIS. As shown as numerical results, BEIS of the convective terms was the most accurate of the two approaches. However, computation was complex. The main objective of this research is to develop a boundary element method for analysing some non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers. On the basis of the boundary element formulation with the pressure penalty function, the divergence theorem is applied to the non-linear convective volume integral of the boundary element formulation. Consequently, velocity gradients are eliminated, the complete formulation is written in terms of velocity. This avoids the difficulty of convective discretizations, and provides considerable reducing storage and computational requirement while improving accuracy. In order to calculate free surface, we construct shooting iterative method of free surface on the basis of normal stress balance condition Project supported by the National Natural Science Foundation of China Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 168 Free and Moving Boundary Problems similar to [2] [3]. The method is applied to analyze two kinds of free surface problems. (1) Solitary wave on free falling thin liquid film. This problem was researched by Wasden and Kheshgi etc. with finite difference method [4] and finite element method [5]. However, in finite difference method[4], one applied an adjustment procedure for wave velocity in order to obtain reasonable flow fields, in finite element methods[5] numerical solution was limited to low flow rates. Moreover, the upwind schemes of discretizing convective momentum terms common to FDM and FEM introduce numerical diffusion or oscillatory behavior of the wave peak region, more importantly, the upwind scheme lacks sensitivity to cross stream diffusion and source terms which are especially important in the case of a thin film. Therefore it is valuable to research a non-traditional numerical method such as BEM. (2) Surface tension drives convective flows in open-boat type crystal growth techniques in microgravity environment. The analytical or numerical study of free surface problem in crystal-growth techniques is very difficult. We shall restrict ourselves to model a problem of open-boat type crystal growth technology, and we only study the melt-gas free boundary, assume the melt-crystal interface shape is flat. The free surface problem of open-boat type is simplified as the free surface problem in open rectangular cavity. The left vertical boundary of the cavity can be interpreted as the crucible wall. The right vertical boundary of the cavity can be interpreted as the melt-crystal interface. The temperatures equal #/ and 0,., respectively. Some important numerical results are presented in paper. BASIC EQUATIONS We consider a system of steady incompressible viscous liquid in a open rectangular cavity with differentially heated lateral walls (the temperature difference is A0*). There is a gradient of surface tension on the free surface since the distribution of temperature is non-homogeneous. The gradient of surface tension drives the convection of liquid. On the basis of the basic law of fluid mechanics and Boussinesq approximation, the governing equations of these problems can be described by a tensor notation and a non-dimensional form as follow: continuity equation «,-„• = o (i) momentum equations energy equation V,-^,=0 (3) where, #,„% is the Kronecker delta symbol, m = x for solitary wave, m = y for thermocapillary convection. The lengths, velocities are scaled with respect to H, Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

A feasibility study has been conducted to assess the applicability of an existing parabolic analysis (ADD-Axisymmetric Diffuser Duct), developed previously for subsonic viscous internal flows, to mixed supersonic/subsonic flows with heat addition simulating a SCRAMJET combustor. A study was conducted with the ADD code modified to include additional convection effects in the normal momentum equation when supersonic expansion and compression waves are present. A set of test problems with weak shock and expansion waves have been analyzed with this modified ADD method and stable and accurate solutions were demonstrated provided the streamwise step size was maintained at levels larger than the boundary layer displacement thickness. Calculations made with further reductions in step size encountered departure solutions consistent with strong interaction theory. Calculations were also performed for a flow field with a flame front in which a specific heat release was imposed to simulate a SCRAMJET combustor. In this case the flame front generated relatively thick shear layers which aggravated the departure solution problem. Qualitatively correct results were obtained for these cases using a marching technique with the convective terms in the normal momentum equation suppressed. It is concluded from the present study that for the class of problems where strong viscous/inviscid interactions are present a global iteration procedure is required.

A three-dimensional unsteady compressible viscous flow solver has been developed, and applied to a low speed centrifugal compressor configuration. Good agreement with experiments have been obtained for a compressor tested at NASA Lewis Research Center. Comparisons with measured surface pressures, and performance data are given. For this compressor, stall was found to occur in the diffuser region. It is demonstrated that this stall may be eliminated, and stable operation may be restored by the use of bleed valves located on the diffuser walls. INTRODUCTION In many army applications (tanks and rotorcraft) small gas turbine engines are often used. The mass flow rate in these engines is small (below 10 Ibs/sec). The compression is provided by one or two centrifugal compressor stages, sometimes in combination with an axial compressor stage. Centrifugal compressors are well suited for handling such low mass flow rates. They are rugged and compact. However, the geometric complexity of the blades and the twisting blade passages give rise to a host of aerodynamic problems such as unsteadiness, establishment of cross flows, flow separation at the trailing edge of the compressor blade, separation over diffuser guide vanes, and unsteady meandering motion of a 'tip vortex' that originates in the vicinity of the gap between the rotor and the casing. In many instances, this unsteadiness leads to rotating stall and surge. Considerable interest exists in the engine community in understanding and eventually controlling these phenomena, particularly compressor stall and surge. Computational fluid dynamics methods provide an efficient way of studying the complex flow phenomena within centrifugal compressors. Thanks to the availability of post-processing and scientific visualization software, the computed flow fields can be extensively analyzed. It is also convenient to test within a numerical model various control strategies such as bleeding, energizing of incoming flow using jets, excitation /wiggling of guide vanes, etc. Passive mechanisms such as slots, slats and flaps can also be easily studied within the scope of a numerical method. These simulations are thus a useful and economical first step to detailed flow control experiments. Finally, CFD simulations provide a wealth of numerical and flow physics information which are useful in developing reduced order models. For these reasons, a 3-D CFD code capable of analyzing centrifugal compressor configurations has been developed by the present investigators. Rather than solving the flow equations in a rotating coordinate system, the present calculations are carried out in an inertial system, with grid velocity terms used to simulate a rotor. Inflow and outflow boundary conditions appropriate for internal flows in general and centrifugal compressors in particular are used. This paper is organized as follows. The flow solution algorithm is first described. Next, results in the form of velocity fields, surface pressure data and performance map are presented for a NASA low-speed centrifugal compressor. The performance of this compressor for off-design low mass flow ratio conditions is subsequently studied. Methods for improving the compressor performance during such conditions are discussed. NUMERICAL FORMULATION The 3-D unsteady compressible Reynolds averaged Navier-Stokes equations are solved numerically using a time marching scheme. This involves solving the governing equations at each time step by marching in time from an initial flow condition with appropriate boundary conditions. The governing equations are cast in a curvilinear body-fitted coordinate system (£,T|,£) and solved using an Alternating Direction Implicit (ADI) scheme 1. This implicit scheme has been validated for a number of fixed wing and rotary wing configurations both in steady and unsteady flight conditions^'*. The partial derivatives in the governing equations are calculated using finite difference approximations. The Graduate Research Assistant, Student Member AIAA. * Regents' professor, Associate Fellow AIAA. Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. inviscid flux terms are calculated using a third order upwind scheme and the viscous stress terms are calculated using standard central difference representations. The details of the upwind formulation are given by Roe. The time derivative is approximated using a two point first order backward difference. The finite difference representation of the governing Navier-Stokes equations at time level 'n' is components at the inflow boundary were set to zero, assuming that there was no swirl at the inlet. It is possible within the framework of the present boundary conditions to prescribe swirl, non-uniform total pressure and total temperature at the inlet. At every time step, the speed of sound a and the normal component of velocity un were found at the inflow boundary from the relations: AT 2a 2a — u = n — u n = I 8 f R + S S + S , T (1) Interior where Aq* is the change in the flow property vector q during adjacent time levels and At is the time step. The operators 8£, 5r| and 8^ are the standard central difference operators. The viscous terms are evaluated explicitly and the inviscid flux vectors F, G and H are calculated implicitly at time level n+1. A three factor approximate factorization is used to factorize the implicit coefficient matrix operator into block tri-diagonal matrices. The factorized system of equations is solved using the Thomas algorithm. Turbulence effects are modeled using the SpalartAllmaras model. INITIAL AND BOUNDARY CONDITIONS The centrifugal compressor is shown in figure 1. The boundary grids are also shown for a single blade passage. The volume grid is made of three segments: inlet region, the rotor, and the diffuser. The following boundary conditions were applied: Periodicity: In the present study, the flow through only a single blade passage has been studied, assuming blade-toblade periodicity. At the periodic boundaries, the flow properties are computed by averaging the properties on either side of the periodic boundary. Solid walls: At the solid walls, the no slip boundary condition was used. The velocity on the walls of the inlet and the diffuser were set to zero, while the velocity for grid points on the compressor blades and the shaft were set equal to Q x r. The pressure, density and temperature values at the solid surfaces were extrapolated from the interior using relationships such as 9p/3n=dT/9n=3p/3n=0, where n represents the direction normal to the solid surface. Inflow Boundary: At this boundary, the stagnation temperature T0 and the total pressure p0 were assumed known. In the simulation of a compressor drawing the air from the atmosphere, these quantities may be set based on ambient conditions. In a multistage compressor analysis, these quantities will be known at the downstream boundary of the previous stage. The tangential (vand w-) (2)

A ’’thin-layer approximation’’ is developed to complement Plesset and Whipple’s [M. S. Plesset and C. G. Whipple, Phys. Fluids 17, 1 (1974)] approximate analysis. Observations of instability of very thin layers reveal that the influence of molecular diffusion may be appreciable.

Implicit schemes are studied with the purpose of introducing them in the two-dimensional GENSMAC method, for the numerical solution of unsteady newtonian incompressible flows. By using the fractional-step approach, the Freeflow2D simulation system is employed to solve the conservation equations. The viscous terms in the Navier-Stokes equations are implicitly treated via the Implicit Backward (IR), Crank-Nicolson (CN) and Adams-Bashforth/Crank-Nicolson (AB/CN) schemes. The convective terms are explicitly treated by the upwind differencing Variable Order Non-Oscillatory Scheme (VONOS) scheme. For channel and impinging jet flows, the numerical techniques are robust and produce results that compare very well with the analytical solutions.

Numerical Investigation of a Darrieus Rotor for Low-head Hydropower Generation

The application of the Kutta condition to unsteady flows has been controversial over the years, with increased research activities over the 1970s and 1980s. This dissatisfaction with the Kutta condition has been recently rejuvenated with the increased interest in low-Reynolds-number, high-frequency bio-inspired flight. However, there is no convincing alternative to the Kutta condition, even though it is not mathematically derived. Realizing that the lift generation and vorticity production are essentially viscous processes, we provide a viscous extension of the classical theory of unsteady aerodynamics by relaxing the Kutta condition. We introduce a trailing-edge singularity term in the pressure distribution and determine its strength by using the triple-deck viscous boundary layer theory. Based on the extended theory, we develop (for the first time) a theoretical viscous (Reynolds-number-dependent) extension of the Theodorsen lift frequency response function. It is found that viscosity induces more phase lag to the Theodorsen function particularly at high frequencies and low Reynolds numbers. The obtained theoretical results are validated against numerical laminar simulations of Navier–Stokes equations over a sinusoidally pitching NACA 0012 at low Reynolds numbers and using Reynolds-averaged Navier–Stokes equations at relatively high Reynolds numbers. The physics behind the observed viscosity-induced lag is discussed in relation to wake viscous damping, circulation development and the Kutta condition. Also, the viscous contribution to the lift is shown to significantly decrease the virtual mass, particularly at high frequencies and Reynolds numbers.

The design of noise-free, low-pollutant combustion devices requires the understanding and the prediction of the combustion noise. The noise originates from the coupling between perturbations in the heat release rate with the oscillating gas (acoustic velocity). Prediction of combustion noise requires the transfer function (TF) that correlates the variation in the heat release rate of the ame with the perturbation (acoustic velocity). Computational costs prohibit an e??cient calculation of the TF by simulating all combustion phenomena associated with the ame dynamics. Simpli??ed theoretical models, such as the hydrodynamic models, allow a signi??cant reduction of the computation time and capture the main features of the ame front dynamics. The fundamental assumption of the hydrodynamic models is that the Bunsen ame, which is described as a thin layer ( ame front) separating the burnt from the unburnt gas, is attached at the burner rim. Within the thin layer approximation, the computation of the TF requires the ame area and, consequenctly, the time-dependent ame front position. By de??ning the ame front as the zero-level set of a distance function, the evolution of the ame front is described by a level set equation. The existent experiments and theoretic kinematic models based on the thin layer approximation (e.g., the G-equation model) provide contradictory information on the phase of the TF. Experiments suggest a low-pass ??lter behaviour of the gain of the TF and an increase of the TF's phase up to several ?? as the frequency increases. In contrast, most of the theoretical models indicate a saturation of the phase of the TF at the level of ??/2. Although the use of a convective wave model for the ow [80] leads to a better agreement between theory and experiments, the origin of the convective wave remains little understood. The discrepancies between theory and experiments most likely reside in the simpli??cations employed in the theoretical models. To understand the behaviour of the experimental TF, here we extend the theoretical models based on the G-equation approach. In the ??rst step, we extend the models by addressing ames that have an arbitrary cone angle and a burning velocity with variable direction. To carry out this ??rst extension we analyse the evolution of a Bunsen ame in a Poiseuille ow from an initially at pro??le to a stationary conical shape. The analysis leads to the correct derivation of the boundary conditions in the extended model employed for the ame evolution in a perturbed Poiseuille ow. We demonstrate that the new model proposed here improves the description of the front close to the boundary and, hence, the behaviour of the ame response to velocity perturbations. Because recent measurements [52] suggest that a better understanding of the TF's phase behaviour requires a hydrodynamic model that accounts for the moving edge of the ame, in the second step we extend the hydrodynamic model to account for non-attached ames. We model the ame front as an open curve by using two level set functions whose zero-level sets are locally normal at the edge point. The movement of the edge along the ame front is accounted for via a novel model for the edge speed. During computer simulations of the ame kinematics with the new extended model the local orthogonality of the level sets is preserved by employing a novel algorithm. The simulations demonstrate that our extended model qualitatively accounts for the ame stabilisation, the ame front and edge kinematics, and the stand-o?? distance of the ame above the burner rim.

On the basis of the thin layer approximation [Ott, Phys. Rev. Lett. 29, 1429 (1972)], a revised thin layer model for incompressible Rayleigh-Taylor instability has been developed to describe the deformation and nonlinear evolution of the perturbed interface. The differential equations for motion are obtained by analyzing the forces (the gravity and pressure difference) of fluid elements (i.e., Newton's second law). The positions of the perturbed interface are obtained from the numerical solution of the motion equations. For the case of vacuum on both sides of the layer, the positions of the upper and lower interfaces obtained from the revised thin layer approximation agree with that from the weakly nonlinear (WN) model of a finite-thickness fluid layer [Wang et al., Phys. Plasmas 21, 122710 (2014)]. For the case considering the fluids on both sides of the layer, the bubble-spike amplitude from the revised thin layer model agrees with that from the WN model [Wang et al., Phys. Plasmas 17, 052305 (2010)] and the expanded Layzer's theory [Goncharov, Phys. Rev. Lett. 88, 134502 (2002)] in the early nonlinear growth regime. Note that the revised thin layer model can be applied to investigate the perturbation growth at arbitrary Atwood numbers. In addition, the large deformation (the large perturbed amplitude and the arbitrary perturbed distributions) in the initial stage can also be described by the present model.

The interaction of vortical structures with a solid wall is studied numerically in this paper. The incompressible Navier–Stokes equations in their primitive variable formulation are solved on a staggered Cartesian grid using the fractional-step algorithm. The convective terms are marched in time using the Adams–Bashforth or Runge–Kutta schemes,while the viscous terms are treated implicitly using the trapezoidal scheme.Anupwind reconstruction of the convective fluxes in their conservative form is proposed and compared against the nonconservative flux formulation, as differences are expected due to nonzero velocity divergence in the numerical solution. The performance of the algorithm is assessed on the convection of an isolated vortex. The algorithm is verified on benchmark incompressible flows and flows involving the convection, mutual interaction, and wall impingement of vortical structures. An attempt is made to simulate the impingement of two and multiple vortex rings against a solid wall to understand the effect of mutual interactions on the interactions with a solid wall. The effects of Reynolds number and initial separation between the rings are studied. These parameters are observed to affect the behavior of the vortical structures, thus affecting the formation and ejection of secondary vortices.

Particle fluid mechanics in shear flows, acoustic waves, and shock waves

An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations

In this article we review recent developments in the analysis and applications of local projection stabilized (LPS) Lagrange–Galerkin (LG) methods to integrate convection dominated-diffusion problems and Navier–Stokes equations at high Reynolds numbers. LG methods combine a discrete Galerkin projection method, usually finite elements, with a backward in time discretization of the convection terms along the characteristic curves. The main advantage of this approach is that provides a natural upwinding to the space discretization of the equations and transform the convection–diffusion problem into an elliptic one (or a Stokes problem in the case of Navier–Stokes equations); however, despite this upwinding introduced by the discretization of the material derivative, LG methods need to be further stabilized. The LPS technique is well suited to stabilize LG methods for the following reasons: (1) it is symmetric and does not break up the symmetry of the discrete system obtained by the backward in time discretization of the material derivative; (2) it is flexible in the sense that can be used with any conventional time marching scheme, and (3) is relatively easy to incorporate in any conventional LG code. In this article, we summarize the last theoretical results concerning the convergence and stability properties when LPS-LG methods are applied with time marching schemes of first and second order.