Discretization Of Navier-Stokes Equations Research Articles

Stability of a continuous/discrete sensitivity model for the Navier–Stokes equations

AbstractThis work presents a comprehensive framework for the sensitivity analysis of the Navier–Stokes equations, with an emphasis on the stability estimate of the discretized first‐order sensitivity of the Navier–Stokes equations. The first‐order sensitivity of the Navier–Stokes equations is defined using the polynomial chaos method, and a finite element‐volume numerical scheme for the Navier–Stokes equations is suggested. This numerical method is integrated into the open‐source industrial code TrioCFD developed by the CEA. The finite element‐volume discretization is extended to the first‐order sensitivity Navier–Stokes equations, and the most significant and original point is the discretization of the nonlinear term. A stability estimate for continuous and discrete Navier–Stokes equations is established. Finally, numerical tests are presented to evaluate the polynomial chaos method and to compare it to the Monte Carlo and Taylor expansion methods.

Positivity-preserving entropy stable schemes for the 1-D compressible Navier-Stokes equations: High-order flux limiting

This is the second of two companion papers, in which we develop and analyze a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 1-D compressible Navier-Stokes equations. The key distinctive property of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method developed in the companion paper [1]. This first-order scheme is discretized on the same Legendre-Gauss-Lobatto points used for the high-order counterpart. As a result, no interpolation or restriction is required between high- and low-order elements which are coupled by using the simultaneous approximation term (SAT) penalty method. To provide positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves the design–order of accuracy of the corresponding high-order baseline spectral collocation method for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorous L2-stability proof for the symmetric form of the discretized Navier-Stokes equations. The proposed methodology is very general and can be extended to multiple dimensions and other summation-by-parts SAT-type schemes.

Distributed deep reinforcement learning for simulation control

Several applications in the scientific simulation of physical systems can be formulated as control/optimization problems. The computational models for such systems generally contain hyperparameters, which control solution fidelity and computational expense. The tuning of these parameters is non-trivial and the general approach is to manually ‘spot-check’ for good combinations. This is because optimal hyperparameter configuration search becomes intractable when the parameter space is large and when they may vary dynamically. To address this issue, we present a framework based on deep reinforcement learning (RL) to train a deep neural network agent that controls a model solve by varying parameters dynamically. First, we validate our RL framework for the problem of controlling chaos in chaotic systems by dynamically changing the parameters of the system. Subsequently, we illustrate the capabilities of our framework for accelerating the convergence of a steady-state computational fluid dynamics solver by automatically adjusting the relaxation factors of the discretized Navier–Stokes equations during run-time. The results indicate that the run-time control of the relaxation factors by the learned policy leads to a significant reduction in the number of iterations for convergence compared to the random selection of the relaxation factors. Our results point to potential benefits for learning adaptive hyperparameter learning strategies across different geometries and boundary conditions with implications for reduced computational campaign expenses 4 4Data and codes available at https://github.com/Romit-Maulik/PAR-RL..

Stabilized dimensional factorization preconditioner for solving incompressible Navier-Stokes equations

In this paper, we propose a stabilized dimensional factorization (SDF) preconditioner for saddle point problems arising from the discretization of Navier-Stokes equations. The idea is based on regularization, block factorization and selective approximation. The spectral properties of the preconditioned matrix are analyzed in details. Based on the analysis, we prescribe a reasonable choice of the regularization matrix W in the preconditioner. By using the connection with the RDF preconditioner, we determine the relaxation parameter α for the problems discretized by uniform grids and stretched grids, respectively. Finally, numerical experiments on the finite element discretizations of both steady and unsteady incompressible flow problems show that the SDF preconditioner is more efficient and robust than the RDF preconditioner, which has been illustrated very competitive with some existing preconditioners.

Flow and heat transfer in a rotating heat pipe with a conical condenser

The flow and heat transfer phenomena in a rotating heat pipe (RHP) with a conical condenser were numerically studied by using a comprehensive numerical model based on the full discretized Navier-Stokes equations. The impact of parameters such as the heat transfer rate, rotational speed, fluid fill charge ratio and working fluid property on the performance of the RHP were examined. The operation characteristics of the RHP was captured in detail by using the present model. It is shown that as the heat transfer rate increased, the temperature gradient along the RHP experienced an increase. The overall thermal resistance of the RHP was reduced due to the enhancement of natural convection in the liquid at the evaporator section with larger heat transfer rates. The RHP has better performance at higher rotational speed in virtue of the enhancement of natural convection induced by larger centrifugal acceleration. The charge of fluid has impact on the performance of the RHP since the thermal resistance of the evaporator experienced an increase with the increase in the fluid fill charge ratio. As the working fluid, water has a better performance than ethanol, due to a larger thermal conductivity, a larger latent heat, and a bigger thermal expansion coefficient.

<i>H</i><sup>2</sup>-stability of some second order fully discrete schemes for the Navier-Stokes equations

This paper considers the $H^2$-stability results for the second order fully discrete schemes based on the mixed finite element method for the 2D time-dependent Navier-Stokes equations with the initial data $u_0∈ H^α, $ where $α = 0, ~1$ and 2. A mixed finite element method is used to the spatial discretization of the Navier-Stokes equations, and the temporal treatments of the spatial discrete Navier-Stokes equations are the second order semi-implicit, implicit/explict and explicit schemes. The $H^2$-stability results of the schemes are provided, where the second order semi-implicit and implicit/explicit schemes are almost unconditionally $H^2$-stable, the second order explicit scheme is conditionally $H^2$-stable in the case of $\alpha = 2$, and the semi-implicit, implicit/explicit and explicit schemes are conditionally $H^2$-stable in the case of $\alpha = 1, ~0$. Finally, some numerical tests are made to verify the above theoretical results.

Applying the free-slip boundary condition with an adaptive Cartesian cut-cell method for complex geometries

The slip condition at the interface of a multiphase flow can occur in situations including micro-and nano-fluidic flow, flow over hydrophobic surfaces, rising bubbles in quiescent liquid, and polymer extrusion processes. The aim of this work is to implement the free-slip boundary condition with an adaptive Cartesian grid method. The Navier-Stokes (NS) equations are solved by a cell-centered collocated finite volume method with adaptive mesh refinement. The arbitrarily-shaped solids imbedded in the computational domain are treated by the cut-cell method where the geometric properties of cut-cells near the boundary are computed through robust geometric operations. In discretized NS equations, the second-order-accurate center difference method is used to estimate the surface fluxes of the regular Cartesian cells in the bulk region, whereas the least-squares method is used to estimate the fluxes of the cut-cells near the boundary. A local coordinate system aligned with the normal and tangential directions of the solid boundary is defined for each cut-cell in order to properly implement the free-slip condition. The tangential velocities at the curved solid boundary are obtained using the free-slip condition and the principal curvatures of the solid surface. The proposed numerical method is implemented in the open-source code Gerris. Numerical tests have been carried out to validate our method. The tests confirm the excellent performances of the proposed method. Although our work focuses on the free-slip condition, the extension of the proposed method to more general slip conditions is straightforward.

Deposition fraction of ellipsoidal fibers in a model of human nasal cavity for laminar and turbulent flows

In this study, the deposition fraction of ellipsoidal fibers in a realistic model of a human nasal cavity was evaluated for laminar and turbulent airflow conditions. The mean flow field was simulated by solving the discretized continuity and Navier-Stokes equations using the ANSYS-Fluent software. For ellipsoidal particle trajectory analysis, several user defined functions (UDFs) were developed and coupled to the ANSYS-Fluent discrete phase model (DPM). The presented formulation accounted for the coupled translation and rotational motions of ellipsoidal fibers and included the stochastic modeling of turbulence velocity and velocity gradient fluctuations. Particular attention was given to the proper modeling of turbulence fluctuation fields especially in the near wall region for accurate evaluation of fiber deposition rate. The predicted fiber deposition fractions in the nasal passage were compared with the available experimental and numerical results for both laminar and turbulent flows and good agreement was observed.

MULTIDIMENSIONAL OPEN SYSTEM FOR VALVELESS PUMPING

In this study, we present a multidimensional open system for valveless pumping (VP). This system consists of an elastic tube connected to two open tanks filled with a fluid under gravity. The two-dimensional elastic tube model is constructed based on the immersed boundary method, and the tank model is governed by a system of ordinary differential equations based on the work-energy principle. The flows into and out of the elastic tube are modeled in terms of the source/sink patches inside the tube. The fluid dynamics of this system is generated by the periodic compress-and-release action applied to an asymmetric region of the elastic tube. We have developed an algorithm to couple these partial differential equations and ordinary differential equations using the pressure-flow relationship and the linearity of the discretized Navier-Stokes equations. We have observed the most important feature of VP, namely, the existence of a unidirectional net flow in the system. Our computations are focused on the factors that strongly influence the occurrence of unidirectional flows, for example, the frequency, compression duration, and location of pumping. Based on these investigations, some case studies are performed to observe the details of the ow features.

Solution of the incompressible Navier–Stokes equations by the method of lines

SummaryThe numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas.The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.

A consistent method for finite volume discretization of body forces on collocated grids applied to flow through an actuator disk

This paper describes a consistent algorithm for eliminating the numerical wiggles appearing when solving the finite volume discretized Navier–Stokes equations with discrete body forces in a collocated grid arrangement. The proposed method is a modification of the Rhie–Chow algorithm where the force in a cell is spread on neighboring cells by applying equivalent pressure jumps at the cell faces. The method shows excellent results when applied for simulating the flow through an actuator disk, which is relevant for wind turbine wake simulations.

Up to sixth-order accurate A-stable implicit schemes applied to the Discontinuous Galerkin discretized Navier–Stokes equations

In this paper a high-order implicit multi-step method, known in the literature as Two Implicit Advanced Step-point (TIAS) method, has been implemented in a high-order Discontinuous Galerkin (DG) solver for the unsteady Euler and Navier–Stokes equations.Application of the absolute stability condition to this class of multi-step multi-stage time discretization methods allowed to determine formulae coefficients which ensure A-stability up to order 6. The stability properties of such schemes have been verified by considering linear model problems. The dispersion and dissipation errors introduced by TIAS method have been investigated by looking at the analytical solution of the oscillation equation.The performance of the high-order accurate, both in space and time, TIAS-DG scheme has been evaluated by computing three test cases: an isentropic convecting vortex under two different testing conditions and a laminar vortex shedding behind a circular cylinder. To illustrate the effectiveness and the advantages of the proposed high-order time discretization, the results of the fourth- and sixth-order accurate TIAS schemes have been compared with the results obtained using the standard second-order accurate Backward Differentiation Formula, BDF2, and the five stage fourth-order accurate Strong Stability Preserving Runge–Kutta scheme, SSPRK4.

Approximate factorization of the discrete Navier-Stokes equations in Cartesian and cylindrical coordinates

A new pseudospectral technique for integrating incompressible Navier-Stokes equations with one nonperiodic boundary in Cartesian or cylindrical coordinate system is presented. Algorithm constructed makes use of Chebyshev collocation technique in nonperiodic direction. Special attention is paid to the approximate factorization of the discrete Navier-Stokes equations in cylindrical geometry leading to highly fast and robust numerical procedure providing spectral accuracy. New approach is an efficient tool for further investigation of turbulent shear flows, for physical hypotheses and alternative algorithms testing. Classical problems of incompressible fluid flows in an infinite plane channel and annuli at transitional Reynolds numbers are taken as model ones.

A Numerical Study on Agglomeration in High Temperature Fluidized beds

Soft-sphere discrete element method (DEM) and Navier-Stokes equations were coupled with equations of energy for gas and solids to investigate the process of agglomeration in fluidized bed of polyethylene particles at high temperature. The Newton’s second law of motion was adapted for translational and rotational motion of particles and agglomerates. The cohesive force for polyethylene particles was calculated based on a time dependent model for solid bridging by the viscous flow mechanism. The motion of agglomerates was described by means of the multi-sphere method. By taking into account the cohesiveness of particles at high temperatures and considering real dynamic agglomerates, the fluidization behavior of a bed of polyethylene particles was successfully simulated in terms of increasing the size of agglomerates. Effect of the inlet gas temperature on mass and size of agglomerates was investigated. A mechanistic study in terms of contact time, cohesive force and repulsive force, which are the key parameters in the formation of agglomerates, were also carried out.

A new stabilized finite element method for shape optimization in the steady Navier–Stokes flow

This paper investigates shape optimization of a solid body located in Navier-Stokes flow in two dimensions. The minimization problem of total dissipated energy is established in the fluid domain. The discretization of Navier-Stokes equations is accomplished using a new stabilized finite element method which does not need a stabilization parameter or calculation of high order derivatives. We derive the structures of discrete Eulerian derivative of the cost functional by a discrete adjoint method with a function space parametrization technique. A gradient type optimization algorithm with a mesh adaptation technique and a mesh moving strategy is effectively formulated and implemented.

Parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with Lagrange multipliers

A parallel approach to solve three-dimensional viscous incompressible fluid flow problems using discontinuous pressure finite elements and a Lagrange multiplier technique is presented. The strategy is based on non-overlapping domain decomposition methods, and Lagrange multipliers are used to enforce continuity at the boundaries between subdomains. The novelty of the work is the coupled approach for solving the velocity–pressure-Lagrange multiplier algebraic system of the discrete Navier–Stokes equations by a distributed memory parallel ILU (0) preconditioned Krylov method. A penalty function on the interface constraints equations is introduced to avoid the failure of the ILU factorization algorithm. To ensure portability of the code, a message based memory distributed model with MPI is employed. The method has been tested over different benchmark cases such as the lid-driven cavity and pipe flow with unstructured tetrahedral grids. It is found that the partition algorithm and the order of the physical variables are central to parallelization performance. A speed-up in the range of 5–13 is obtained with 16 processors. Finally, the algorithm is tested over an industrial case using up to 128 processors. In considering the literature, the obtained speed-ups on distributed and shared memory computers are found very competitive.

Building a reduced model for nonlinear dynamics in Rayleigh-Bénard convection with counter-rotating disks

A reduced model to decrease the number of degrees of freedom of the discretized Navier-Stokes equations to a small set that nevertheless captures the essential dynamics of the flow is proposed. The Rayleigh-Bénard convection problem in a cylinder of aspect ratio one where the lower and upper disks, maintained at hot and cold temperatures, respectively, rotate at equal and opposite angular velocities has been chosen to test the technique. The nonlinear dynamics is rich and complex when the temperature difference between disks and their angular velocity is varied. Representatives states--stationary, periodic near sinusoidal, and near heteroclinic--are presented. In each case, the reduced model is compared with temporal integration, and we show that 41 degrees of freedom are sufficient to reproduce the signal. We discuss the strengths and weaknesses of the algorithm by which we build our reduced model.

Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier–Stokes equations

The fluid dynamic equations are discretized by a high-order spectral volume (SV) method on unstructured tetrahedral grids. We solve the steady state equations by advancing in time using a backward Euler (BE) scheme. To avoid the inversion of a large matrix we approximate BE by an implicit lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm. The implicit method addresses the stiffness in the discrete Navier–Stokes equations associated with stretched meshes. The LU-SGS algorithm is then used as a smoother for a p-multigrid approach. A Von Neumann stability analysis is applied to the two-dimensional linear advection equation to determine its damping properties. The implicit LU-SGS scheme is used to solve the two-dimensional (2D) compressible laminar Navier–Stokes equations. We compute the solution of a laminar external flow over a cylinder and around an airfoil at low Mach number. We compare the convergence rates with explicit Runge–Kutta (E-RK) schemes employed as a smoother. The effects of the cell aspect ratio and the low Mach number on the convergence are investigated. With the p-multigrid method and the implicit smoother the computational time can be reduced by a factor of up to 5–10 compared with a well tuned E-RK scheme.

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.

Direct numerical simulation of sensitivity coefficients in low Reynolds number turbulent channel flow

The sensitivity equation method (SEM) has been implemented in direct numerical simulations (DNS) of smooth-wall turbulent channel flow to quantify Reynolds number effects on the mean flow field. The present approach towards sensitivity analysis represents a departure from conventional parametric studies wherein multiple numerical simulations are performed with discrete increments in the value of the parameter of interest, in this case the Reynolds number. In the present study, sensitivity derivatives (or sensitivity coefficients) represent the rate of change of the primitive variables with respect to the Reynolds number, and are determined directly by numerically solving the discretized continuous sensitivity equations concurrently with the discretized Navier–Stokes equations. Simulations have been performed at Reynolds numbers of 100 and 180, based on the friction velocity and channel half-width, using a finite volume computational scheme. Wall-normal profiles of the mean streamwise velocity and Reynolds stresses compare very well to others in the literature at the same Reynolds numbers [19,20]. The SEM results correctly predict the expected change in both the mean streamwise velocity and Reynolds shear stress profiles with increasing Reynolds number. Furthermore, the mean SEM results correctly predict the local slope of the skin friction coefficient versus Reynolds number curve, which may have potential use in applications such as boundary layer control by surface modification. The present study is novel insofar as it constitutes the first attempt to implement SEM in an unsteady, fully turbulent, three-dimensional flow field. The additional computational expenses incurred in order to run the SEM simulations in this context are also discussed.