Generalization of the RANS Equations Using Mean Modal Decomposition of the Navier Stokes Equations

Volume 1 ◽  
2004 ◽  
Author(s):  
Fereydoun Sabetghadam
Keyword(s):  
Stokes Equations ◽  
Order System ◽  
Navier Stokes ◽  
Real Field ◽  
Navier Stokes Equations ◽  
Rans Equations ◽  
Zeroth Mode ◽  
Reynolds Decomposition ◽  
Coherent Vortex ◽  
Part Decomposition

A generalization in the Reynolds decomposition and averaging are proposed in this paper. The method is directly applied to the Navier Stokes (N-S) equations to construction of a generalized Reynolds Averaged Navier Stokes (RANS) equations. The formulation which is presented for the fields realized in a suitable ensemble, is based on a two part decomposition. One part is an approximate unique representation of the field and when reconstruction of the field, will repeat in all ensemble elements. The other part represents deviation of the real field from the approximate part and therefore is different in any mode and each ensemble element. The decomposition is applied in both spatial and temporal fashions. In the temporal decomposition, a system of Partial Differential Equations (PDEs) is obtained that is nonclosed, coupled and second order in space and its zeroth mode is the classical Reynolds averaged values of the field. In the spatial decomposition whereas, a first order system of nonclosed PDEs is obtained which could be seen as an alternative version of the Proper Orthogonal Decomposition (POD) or the Coherent Vortex Simulation (CVS) methods. In both fashions however, there are some terms that must be modeled just like as the classical closure problem in the RANS method. The method is applied on a one dimensional mixed random-nonrandom field and successfully extracted the coherent part of the field.

Modeling coherent structures in the atmosphere and assessing their impact on aircraft

2021 ◽  
Author(s):  
V.V. Vyshinsky ◽  
K.T. Zoan
Keyword(s):  
Stokes Equations ◽  
Grid Method ◽  
Computer Code ◽  
Light Transport ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Large Structures ◽  
Glide Path ◽  
Coherent Vortex ◽  
Coherent Vortex Structure

The paper introduces an engineering method for assessing the aerodynamic effect of disturbed atmosphere on an aircraft. As a source of vortex structures, we can consider vortex wind wakes that arise when the atmospheric wind flows around the landscape, large structures, moving or stationary aircraft-carrying platforms, vortex wakes behind aircraft, etc. In this study, we consider the situation when a light transport aircraft and an aircraft of the MC-21 type get into the vortex wake behind the super-heavy aircraft A-380 when flying along the glide path. A coherent vortex structure behind the A-380 is formed by the grid method within the framework of the boundary value problem for the Reynolds-averaged Navier —Stokes equations. The evolution and stochastics of the far wake are carried out using the author’s computer code written in the MATLAB system, within the framework of discrete vortices with a Rankine core. The assessment of the increment of forces and moments from the effect of the vortex system on the aircraft was carried out using the panel method.

scholarly journals Thresholds for the formation of satellites in two-dimensional vortices

2008 ◽  
Vol 614 ◽  
pp. 381-405 ◽  
Author(s):  
M. R. TURNER ◽  
A. D. GILBERT
Keyword(s):  
Normal Mode ◽  
Stokes Equations ◽  
Critical Layer ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Coherent Vortex ◽  
Neutral Mode ◽  
The Mean ◽  
Fourier Harmonics

This paper examines the evolution of a two-dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m = 2 added to it. If the perturbation is weak, then the vortex returns to an axisymmetric state and the non-zero Fourier harmonics generated by the perturbation decay to zero. However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it.The paper considers the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine-scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative diagnostics, the appearance of an inflection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical-layer equations numerically. These calculations are supported by simulations of the full Navier–Stokes equations using a family of profiles based on the tanh function.

scholarly journals Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15
Author(s):  
Nejib Smaoui
Keyword(s):  
Numerical Simulations ◽  
Stokes Equations ◽  
Order System ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Number Range ◽  
Kolmogorov Flow ◽  
Dynamics And Control ◽  
Seventh Order ◽  
And Control

The symmetries, dynamics, and control problem of the two-dimensional (2D) Kolmogorov flow are addressed. The 2D Kolmogorov flow is known as the 2D Navier-Stokes (N-S) equations with periodic boundary conditions and with a sinusoidal external force along the x-direction. First, using the Fourier Galerkin method on the original 2D Navier-Stokes equations, we obtain a seventh-order system of nonlinear ordinary differential equations (ODEs) which approximates the behavior of the Kolmogorov flow. The dynamics and symmetries of the reduced seventh-order ODE system are analyzed through computer simulations for the Reynolds number range 0<Re<26.41. Extensive numerical simulations show that the obtained system is able to display the different behaviors of the Kolmogorov flow. Then, we design Lyapunov based controllers to control the dynamics of the system of ODEs to different attractors (e.g., a fixed point, a periodic orbit, or a chaotic attractor). Finally, numerical simulations are undertaken to validate the theoretical developments.

scholarly journals Causal dissipation for the relativistic dynamics of ideal gases

2017 ◽  
Vol 473 (2201) ◽  
pp. 20160729 ◽  
Author(s):  
Heinrich Freistühler ◽  
Blake Temple
Keyword(s):  
Euler Equations ◽  
Stokes Equations ◽  
Second Order ◽  
Order System ◽  
Navier Stokes ◽  
Unique Element ◽  
Relativistic Euler Equations ◽  
Navier Stokes Equations ◽  
First Order ◽  
Ideal Gases

We derive a general class of relativistic dissipation tensors by requiring that, combined with the relativistic Euler equations, they form a second-order system of partial differential equations which is symmetric hyperbolic in a second-order sense when written in the natural Godunov variables that make the Euler equations symmetric hyperbolic in the first-order sense. We show that this class contains a unique element representing a causal formulation of relativistic dissipative fluid dynamics which (i) is equivalent to the classical descriptions by Eckart and Landau to first order in the coefficients of viscosity and heat conduction and (ii) has its signal speeds bounded sharply by the speed of light. Based on these properties, we propose this system as a natural candidate for the relativistic counterpart of the classical Navier–Stokes equations.

A Symmetric Direct Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations

2017 ◽  
Vol 22 (2) ◽  
pp. 375-392 ◽  
Author(s):  
Huiqiang Yue ◽  
Jian Cheng ◽  
Tiegang Liu
Keyword(s):  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Equivalent Form ◽  
Second Order ◽  
Heat Fluxes ◽  
Order System ◽  
Navier Stokes ◽  
Optimal Order ◽  
Navier Stokes Equations ◽  
Optimal Order Of Convergence

AbstractIn this work, we investigate the numerical approximation of the compressible Navier-Stokes equations under the framework of discontinuous Galerkin methods. For discretization of the viscous and heat fluxes, we extend and apply the symmetric direct discontinuous Galerkin (SDDG) method which is originally introduced for scalar diffusion problems. The original compressible Navier-Stokes equations are rewritten into an equivalent form via homogeneity tensors. Then, the numerical diffusive fluxes are constructed from the weak formulation of primal equations directly without converting the second-order equations to a first-order system. Additional numerical flux functions involving the jump of second order derivative of test functions are added to the original direct discontinuous Galerkin (DDG) discretization. A number of numerical tests are carried out to assess the practical performance of the SDDG method for the two dimensional compressible Navier-Stokes equations. These numerical results obtained demonstrate that the SDDG method can achieve the optimal order of accuracy. Especially, compared with the well-established symmetric interior penalty (SIP) method [18], the SDDG method can maintain the expected optimal order of convergence with a smaller penalty coefficient.

Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

2020 ◽  
Vol 14 (4) ◽  
pp. 7369-7378
Author(s):  
Ky-Quang Pham ◽  
Xuan-Truong Le ◽  
Cong-Truong Dinh
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Aerodynamic Performance ◽  
Numerical Results ◽  
Single Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Operating Stability ◽  
Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

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