An Implicit Multigrid Scheme for the Compressible Navier-Stokes Equations With Low-Reynolds-Number Turbulence Closure

1998 ◽  
Vol 120 (2) ◽  
pp. 257-262 ◽  
Author(s):  
Peter Gerlinger ◽  
Dieter Bru¨ggemann
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Convergence Acceleration ◽  
Iterative Solvers ◽  
Turbulence Closure ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Reynolds ◽  
Turbulence Transport

A multigrid method for convergence acceleration is used for solving coupled fluid and turbulence transport equations. For turbulence closure a low-Reynolds-number q-ω turbulence model is employed, which requires very fine grids in the near wall regions. Due to the use of fine grids, convergence of most iterative solvers slows down, making the use of multigrid techniques especially attractive. However, special care has to be taken on the strong nonlinear turbulent source terms during restriction from fine to coarse grids. Due to the hyperbolic character of the governing equations in supersonic flows and the occurrence of shock waves, modifications to standard multigrid techniques are necessary. A simple and effective method is presented that enables the multigrid scheme to converge. A strong reduction in the required number of multigrid cycles and work units is achieved for different test cases, including a Mack 2 flow over a backward facing step.

Drag Force on a Sphere Moving in Low-Reynolds-Number Pipe Flows

2007 ◽  
Vol 23 (4) ◽  
pp. 423-432 ◽  
Author(s):  
S.-H. Lee ◽  
Tzuyin Wu
Keyword(s):  
Reynolds Number ◽  
Drag Force ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Time Derivative ◽  
Creeping Flow ◽  
Navier Stokes ◽  
Sphere Diameter ◽  
Navier Stokes Equations ◽  
Low Reynolds

AbstractIn this paper, the drag force on a sphere moving constantly along the centerline of a circular pipe filled with viscous fluid (the falling-sphere problem) under low Reynolds number condition is investigated via numerical calculation. The incompressible Navier-Stokes equations are formulated in a pseudocompressibility form. The numerical scheme makes use of finite-volume method and the numerical flux terms are evaluated using the Total-Variation Diminishing (TVD) strategy commonly applied to the compressible flow. Steady solution is obtained by marching (iterating) in time until the artificial time derivative of pressure term in the continuity equation drops to zero.In the calculation, six different Reynolds number (Re) ranging from 0.1 to 1 and seven different pipe-to-sphere diameter ratios (D/d) ranging from 5 to 40 are selected to study the pipe-wall effect. In each case, the drag force on the sphere is evaluated and the results are compared with the existing approximate theoretical values derived from correcting the Stokes' formula. Both results agree in trend, but with noticeable deviation in values, particularly for cases with large pipe-to-sphere diameter ratios. The deviation is due to the fact that theoretical values were based on the solution to the linearized Navier-Stokes equations (Stokes' creeping-flow equations), while the fully nonlinear form of the Navier-Stokes equations are adopted in the present calculations. Finally, a least-square regression technique is applied to collapse the calculated results into a single expression exhibiting the functional relationship between the drag force, Reynolds number (Re), and the pipe-to-sphere diameter ratio (D/d).

Numerical Simulation of a Low Reynolds Number Airfoil

2013 ◽  
Vol 390 ◽  
pp. 141-146
Author(s):  
Yu Fu Wang ◽  
Guo Quan Tao ◽  
Ze Hai Wang ◽  
Zhe Wu
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Lift Coefficient ◽  
Aerodynamic Characteristics ◽  
Navier Stokes ◽  
Complex Flow ◽  
Navier Stokes Equations ◽  
Low Reynolds ◽  
Low Reynolds Number Airfoil

In this paper, a low Reynolds number airfoil (S1223) is the objective of the study. The Navier-Stokes equations were established to simulate the complex flow around a low Reynolds number airfoil, in which the turbulence model was used. The complex flow around the airfoil was simulated at 2x105 Reynolds number and its aerodynamic characteristics were analyzed. The relationship among lift coefficient, drag coefficient and angle of attack was studied.

Viscous extension of potential-flow unsteady aerodynamics: the lift frequency response problem

2019 ◽  
Vol 868 ◽  
pp. 141-175 ◽  
Author(s):  
Haithem Taha ◽  
Amir S. Rezaei
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Unsteady Aerodynamics ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Reynolds Numbers ◽  
High Frequencies ◽  
Kutta Condition ◽  
Low Reynolds

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.

Influence of Unsteady Parameters on the Aerodynamics of a Low Reynolds Number Pitching Airfoil

2009 ◽  
Author(s):  
M. R. Amiralaei ◽  
H. Alighanbari ◽  
S. M. Hashemi
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Analytical Data ◽  
Hysteresis Loops ◽  
Aerodynamic Performance ◽  
Navier Stokes ◽  
Pitching Airfoil ◽  
Naca0012 Airfoil ◽  
Low Reynolds

The objective of the present study is to investigate low Reynolds number aerodynamics of a harmonically pitching NACA0012 airfoil. To this mean, the influence of some unsteady parameters; amplitude of oscillation, d, reduced frequency, k, and Reynolds number, Re, on the aerodynamic performance of the airfoil is investigated. Computational Fluid Dynamics (CFD) is utilized to solve Navier-Stokes equations discretized based on Finite Volume Method (FVM). The instantaneous lift coefficients are obtained and compared with analytical data from Theodorsen’s equations. The simulation results reveal that d, k, and Re are of great importance in the aerodynamic performance. They affect the maximum lift coefficients, hysteresis loops, strength and number of generated vortices within the harmonic motion, and the extent of the figure-eight phenomenon region. Thus, the optimum aerodynamic performance demands a careful selection of these parameters.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

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