Showing papers on "Navier–Stokes equations published in 2002"

Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics

Abstract: Error estimates for Galerkin proper orthogonal decomposition (POD) methods for nonlinear parabolic systems arising in fluid dynamics are proved For the time integration the backward Euler scheme is considered The asymptotic estimates involve the singular values of the POD snapshot set and the grid-structure of the time discretization as well as the snapshot locations

Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique

Abstract: A harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented. The analysis exploits the fact that many unsteady e ows of interest in turbomachinery are periodic in time. Thus, the unsteady e ow conservation variables may be represented by a Fourier series in time with spatially varying coefe cients. This assumption leads to a harmonic balance form of the Euler or Navier ‐Stokes equations, which, in turn, can be solved efe ciently as a steady problem using conventional computational e uid dynamic (CFD) methods, including pseudotime time marching with local time stepping and multigrid acceleration. Thus, the method is computationally efe cient, at least one to two orders of magnitude faster than conventional nonlinear time-domain CFD simulations. Computational results for unsteady, transonic, viscous e ow in the front stage rotor of a high-pressure compressor demonstrate that even strongly nonlinear e ows can be modeled to engineering accuracy with a small number of terms retained in the Fourier series representation of the e ow. Furthermore, in some cases, e uid nonlinearities are found to be important for surprisingly small blade vibrations.

Heat transfer in stagnation-point flow towards a stretching sheet

Abstract: Steady two-dimensional stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that for a fluid of small kinematic viscosity, a boundary layer is formed when the stretching velocity is less than the free stream velocity and an inverted boundary layer is formed when the stretching velocity exceeds the free stream velocity. Temperature distribution in the boundary layer is found when the surface is held at constant temperature and surface heat flux is determined.

The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory

Abstract: We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokes-alpha) (NS-α) equations. We also provide estimates, in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the Kolmogorov theory of turbulence, we define a small spatial scale, l ∈ , as the scale at which the balance occurs in the mean rates of nonlinear transport of energy and viscous dissipation of energy. Furthermore, we show that the number of degrees of freedom in the long-time behavior of the solutions to these equations is bounded from above by (L/l ∈ )3, where L is a typical large spatial scale (e.g., the size of the domain). This estimate suggests that the Landau–Lifshitz classical theory of turbulence is suitable for interpreting the solutions of the NS-α equations. Hence, one may consider these equations as a closure model for the Reynolds averaged Navier–Stokes equations (NSE). We study this approach, further, in other related papers. Finally, we discuss the relation of the NS-α model to the NSE by proving a convergence theorem, that as the length scale α 1 tends to zero a subsequence of solutions of the NS-α equations converges to a weak solution of the three dimensional NSE.

Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation

Abstract: Velocity field statistics in the inertial to dissipation range of three-dimensional homogeneous steady turbulent flow are studied using a high-resolution DNS with up to N=10243 grid points. The range of the Taylor microscale Reynolds number is between 38 and 460. Isotropy at the small scales of motion is well satisfied from half the integral scale (L) down to the Kolmogorov scale (η). The Kolmogorov constant is 1.64±0.04, which is close to experimentally determined values. The third order moment of the longitudinal velocity difference scales as the separation distance r, and its coefficient is close to 4/5. A clear inertial range is observed for moments of the velocity difference up to the tenth order, between 2λ≈100η and L/2≈300η, where λ is the Taylor microscale. The scaling exponents are measured directly from the structure functions; the transverse scaling exponents are smaller than the longitudinal exponents when the order is greater than four. The crossover length of the longitudinal velocity struct...

Dynamic wall modeling for large-eddy simulation of complex turbulent flows

Abstract: The efficacy of large-eddy simulation (LES) with wall modeling for complex turbulent flows is assessed by considering turbulent boundary-layer flows past an asymmetric trailing-edge. Wall models based on turbulent boundary-layer equations and their simpler variants are employed to compute the instantaneous wall shear stress, which is used as approximate boundary conditions for the LES. It is demonstrated that, as first noted by Cabot and Moin [Flow Turb. Combust. 63, 269 (2000)], when a Reynolds-averaged Navier–Stokes type eddy viscosity is used in the wall-layer equations with nonlinear convective terms, its value must be reduced to account for only the unresolved part of the Reynolds stress. A dynamically adjusted mixing-length eddy viscosity is used in the turbulent boundary-layer equation model, which is shown to be considerably more accurate than the simpler wall models based on the instantaneous log law. This method predicts low-order velocity statistics in good agreement with those from the full LES with resolved wall-layers, at a small fraction of the original computational cost. In particular, the unsteady separation near the trailing-edge is captured correctly, and the prediction of surface pressure fluctuations also shows promise.

An implicit velocity decoupling procedure for the incompressible Navier–Stokes equations

Abstract: An efficient numerical method to solve the unsteady incompressible Navier-Stokes equations is developed. A fully implicit time advancement is employed to avoid the Courant-Friedrichs-Lewy restriction, where the Crank-Nicolson discretization is used for both the diffusion and convection terms, Based on a block LU decomposition, velocity -pressure decoupling is achieved in conjunction with the approximate factorization The main emphasis is placed on the additional decoupling of the intermediate velocity components with only nth time step velocity. The temporal second-order accuracy is preserved with the approximate factorization without any modification of boundary conditions. Since the decoupled momentum equations are solved without iteration, the computational time is reduced significantly. The present decoupling method is validated by solving several test cases in particular, the turbulent minimal channel flow unit

A Preconditioner for the Steady-State Navier--Stokes Equations

Abstract: We present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady-state Navier--Stokes equations (also known as the Oseen equations). The solver is an iterative method of Krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the Oseen operator. The preconditioner may also be conceived through a weaker heuristic argument involving differential operators. Computations indicate that convergence for the preconditioned discrete Oseen problem is only mildly dependent on the viscosity (inverse Reynolds number) and, most importantly, that the number of iterations does not grow as the mesh size is reduced. Indeed, since the preconditioner is motivated through analysis of continuous operators, the number of iterations decreases for smaller mesh size which accords with better approximation of these operators.

Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R2

Abstract: We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R 2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows us to extend those results in a number of ways.

Numerical Treatment of Defective Boundary Conditions for the Navier--Stokes Equations

Abstract: We present a formulation for accommodating defective boundary conditions for the incompressible Navier--Stokes equations where only averaged values are prescribed on measurable portions of the boundary. In particular we consider the case where the flow rate is imposed on several domain sections. This methodology has an interesting application in the numerical simulation of flow in blood vessels, when only a reduced set of boundary data are generally available for the upstream and downstream sections.

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Abstract: We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers.

Dynamics of homogeneous bubbly flows Part 1. Rise velocity and microstructure of the bubbles

Abstract: Direct numerical simulations of the motion of up to 216 three-dimensional buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a parallelized finite-difference/front-tracking method that allows a deformable interface between the bubbles and the suspending fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is about 12–30, depending on the void fraction; deformations of the bubbles are small. Although the motion of the individual bubbles is unsteady, the simulations are carried out for a sufficient time that the average behaviour of the system is well defined. Simulations with different numbers of bubbles are used to explore the dependence of the statistical quantities on the size of the system. Examination of the microstructure of the bubbles reveals that the bubbles are dispersed approximately homogeneously through the flow field and that pairs of bubbles tend to align horizontally. The dependence of the statistical properties of the flow on the void fraction is analysed. The dispersion of the bubbles and the fluctuation characteristics, or ‘pseudo-turbulence’, of the liquid phase are examined in Part 2.

Exponential Convergence for the Stochastically Forced Navier-Stokes Equations and Other Partially Dissipative Dynamics

Abstract: We prove that the two dimensional Navier-Stokes equations possess an exponentially attracting invariant measure. This result is in fact the consequence of a more general ``Harris-like'' ergodic theorem applicable to many dissipative stochastic PDEs and stochastic processes with memory. A simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting. To analyze the iterated map, a general ``Doeblin-like'' theorem is proven. One of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non-Markovian processes.

Energy dissipation in body-forced turbulence

Abstract: Bounds on the bulk rate of energy dissipation in body-force-driven steady-state turbulence are derived directly from the incompressible Navier–Stokes equations We consider flows in three spatial dimensions in the absence of boundaries and derive rigorous a priori estimates for the time-averaged energy dissipation rate per unit mass, e, without making any further assumptions on the flows or turbulent fluctuations We provee [les ] c1v U2/l2 + c2 U3/l,where v is the kinematic viscosity, U is the root-mean-square (space and time averaged) velocity, and l is the longest length scale in the applied forcing function The prefactors c1 and c2 depend only on the functional shape of the body force and not on its magnitude or any other length scales in the force, the domain or the flow We also derive a new lower bound on e in terms of the magnitude of the driving force F For large Grashof number Gr = Fl3/v2, we findc3 vFl/λ2 [les ] ewhere λ = √vU2/e is the Taylor microscale in the flow and the coefficient c3 depends only on the shape of the body force This estimate is seen to be sharp for particular forcing functions producing steady flows with λ/l ∼ O(1) as Gr → 1 We interpret both the upper and lower bounds on e in terms of the conventional scaling theory of turbulence – where they are seen to be saturated – and discuss them in the context of experiments and direct numerical simulations

Computation of trailing-edge noise due to turbulent flow over an airfoil

Abstract: Application of the variational formulation of Lighthill’ s acoustic analogy to trailing-edge noise is considered. Use is made of this formulation to study the effect of e niteness of the chord and the variation of far-e eld pressure directivity with frequency. Numerical analytical solution results are validated for certain limiting cases. Use is also made of this methodology to calculate the far-e eld acoustic pressure for a low-Mach-number turbulent e ow. To determine the acoustic sources for this problem, we employ an unstructured mesh, large eddy simulation of the incompressible Navier ‐Stokes equations.

Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations

Abstract: In this paper we consider the Cauchy problem for the n-dimensional Navier-Stokes equations and we prove a regularity criterion for weak solutions involving the summability of the pressure. Related results for the initial-boundary value problem are also presented.

Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier–Stokes equations

Abstract: We present a critical comparison between two recently proposed discontinuous Galerkin methods for the space discretization of the viscous terms of the compressible Navier-Stokes equations. The robustness and accuracy of the two methods has been numerically evaluated by considering simple but well documented classical two-dimensional test cases, including the flow around the NACA0012 airfoil, the flow along a flat plate and the flow through a turbine nozzle

A class of higher order compact schemes for the unsteady two‐dimensional convection–diffusion equation with variable convection coefficients

Abstract: A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two-dimensional unsteady convection-diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter μ and fourth order accurate in space. For 0.5 ≤ μ ≤ 1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection-diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection-diffusion problem and three flows of varying complexities governed by the two-dimensional incompressible Navier-Stokes equations

Dynamic One-Equation Nonviscosity Large-Eddy Simulation Model

Abstract: Anew approach for a nonviscosity large-eddy simulation (LES)subgrid stress model is presented.Theapproach uses a scaling that is provided by the subgrid kinetic energy and a tensor coefe cient that is obtained from the dynamic modeling approach, hence, a dynamic structure model. Mathematical and conceptual issues motivating the development of this new model are explored. Attention is focused on dynamic modeling approaches. The basic equations that originate in dynamic modeling approaches are Fredholm integral equations of the second kind. These equations have solvability requirements that have not been previously addressed in the context of LES models. These conditionsare examined for traditional dynamic Smagorinksy modeling, that is, zero-equation approaches, and one-equation subgrid models. It is shown that standard approaches do not always satisfy the integral equation solvability condition. It is also shown that traditional LES models that use the resolved scale strain rate to estimatethesubgrid stressesscalepoorly with e lterlevel, leading to signie cant errorsin themodeling of the subgrid scale stress. The poor scaling in traditional LES approaches can result in not only weak models, but can also cause nonrealizability of the subgrid stresses. A better scaling based on the subgrid kinetic energy is proposed that leads to a new one-equation nonviscosity model that does satisfy the solvability conditions and appears to maintain realizability. Both integral and algebraic formulations of the new one-equation nonviscosity model are presented. The resolved and subgrid kinetic energies are shown to compare well to a direct numerical simulation of decaying isotropic turbulence.

Computational aeroacoustics of phonation, part I: Computational methods and sound generation mechanisms.

Abstract: The aerodynamic generation of sound during phonation was studied using direct numerical simulations of the airflow and the sound field in a rigid pipe with a modulated orifice. Forced oscillations with an imposed wall motion were considered, neglecting fluid–structure interactions. The compressible, two-dimensional, axisymmetric form of the Navier–Stokes equations were numerically integrated using highly accurate finite difference methods. A moving grid was used to model the effects of the moving walls. The geometry and flow conditions were selected to approximate the flow within an idealized human glottis and vocal tract during phonation. Direct simulations of the flow and farfield sound were performed for several wall motion programs, and flow conditions. An acoustic analogy based on the Ffowcs Williams–Hawkings equation was then used to decompose the acoustic source into its monopole, dipole, and quadrupole contributions for analysis. The predictions of the farfield acoustic pressure using the acoustic...

Large-Eddy Simulation of the Flow Around a Bluff Body

Abstract: Large-eddy simulations are made of the flow around a surface-mounted cube, showing that it is possible to obtain accurate results in a coarse grid simulation. The inadequate resolution is compensated for by the use of a dynamic one-equation subgrid-scale model. Two one-equation subgrid models are used here to model the subgrid-scale stress tensor. A series of time-averaged velocities and turbulent stresses are computed and compared with the experiments. Global quantities such as drag and lift coefficients and vortex shedding frequency are presented. The transfer of the turbulent energy was studied and the reverse transfer of energy (backscatter) was predicted. Coherent structures and other flow features were also examined

Newton-Krylov Algorithm for Aerodynamic Design Using the Navier-Stokes Equations

Abstract: A Newton‐Krylov algorithm is presented for two-dimensional Navier‐Stokes aerodynamic shape optimization problems. The algorithm is applied to both the discrete-adjoint and the discrete e ow-sensitivity methods for calculating the gradient of the objective function. The adjoint and e ow-sensitivity equations are solved using a novel preconditioned generalized minimum residual (GMRES)strategy. Together with a complete linearization of the discretized Navier‐Stokes and turbulence model equations, this results in an accurate and efecient evaluation of the gradient. Furthermore, fast e ow solutions are obtained using the same preconditioned GMRES strategy in conjunction with an inexact Newton approach. The performance of the new algorithm is demonstrated for several design examples,includinginversedesign,lift-constraineddragminimization, liftenhancement, and maximization of lift-to-dragratio. In all examples, the normof the gradientisreduced by several ordersof magnitude, indicating that alocalminimumhasbeen obtained. Bytheuseoftheadjoint method,thegradient isobtained infromone-e fth to one-half of the time required to converge a eow solution.