Energy dissipation in body-forced turbulence

2002 ◽  
Vol 467 ◽  
pp. 289-306 ◽  
Author(s):  
CHARLES R. DOERING ◽  
CIPRIAN FOIAS
Keyword(s):  
Energy Dissipation ◽  
Body Force ◽  
Stokes Equations ◽  
A Priori ◽  
Energy Dissipation Rate ◽  
The Body ◽  
Upper And Lower Bounds ◽  
Navier Stokes ◽  
Steady Flows ◽  
Forcing Function

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, ε, without making any further assumptions on the flows or turbulent fluctuations. We proveε [les ] c1vU2/l2 + c2U3/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 ε in terms of the magnitude of the driving force F. For large Grashof number Gr = Fl3/v2, we findc3vFl/λ2 [les ] εwhere λ = √vU2/ε 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 ε 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.

Comparison of Different Modeling Techniques for Solid-Fluid Interface for Wave Loading

2021 ◽  
Author(s):  
Tara Saladin ◽  
Young W. Kwon ◽  
Joseph T. Klamo
Keyword(s):  
Energy Dissipation ◽  
Wave Energy ◽  
Stokes Equations ◽  
The Body ◽  
Navier Stokes ◽  
Wave Loading ◽  
Wave Energy Dissipation ◽  
Submerged Body ◽  
Modeling Techniques ◽  
Fluid Solid Interaction

Abstract Computational fluid dynamics (CFD) has been used to estimate the wave loading applied to a fully submerged body near the surface. The Navier-Stokes equations were used for the present study. In terms of modeling the fluid-solid interface, two different techniques are available in ANSYS CFX. One is the Rigid Body Method (RBM) and the other is the Immersed Solid Method (ISM). This paper compares the two modeling techniques in terms of accuracy and modeling flexibility. For this study, a CFD model of the NPS tow tank with wave generation and a submerged body was created to investigate different methods of solid body modeling. A comparison of the RBM and ISM was performed modeling a submerged rectangular body at different depths. The models produced similar results when the body was lower beneath the wave surface with limited fluid-solid interaction. As the amount of fluid-solid interaction increased, the RBM showed increased amounts of wave energy dissipation as compared to the ISM. This disruption of the wave energy resulted in the RBM showing smaller body forces and moments when compared to the ISM solid model. The increased wave energy dissipation in the RBM is likely caused by the different mechanism for modeling body-solid interaction. The numerical results were also compared to the experimental data.

scholarly journals Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces

2019 ◽  
Vol 150 (2) ◽  
pp. 569-606 ◽  
Author(s):  
Dat Cao ◽  
Luan Hoang
Keyword(s):  
Asymptotic Expansion ◽  
Body Force ◽  
Stokes Equations ◽  
Three Dimensional ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Gevrey Space ◽  
Long Time ◽  
The Individual

AbstractThe Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

scholarly journals A Class of New Exact Solutions of Navier-Stokes Equations with Body Force For Viscous Incompressible Fluid

2018 ◽  
Vol 7 (1) ◽  
pp. 20
Author(s):  
Mushtaq Ahmed ◽  
Rana Khalid Naeem ◽  
Syed Anwer Ali
Keyword(s):  
Incompressible Fluid ◽  
Exact Solutions ◽  
Body Force ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Steady Motion ◽  
The Body ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
New Exact Solutions

This paper is to indicate a class of new exact solutions of the equations governing the two-dimensional steady motion of incompressible fluid of variable viscosity in the presence of body force. The class consists of the stream function $\psi$ characterized by equation $\theta=f(r)+ a \psi + b $ in polar coordinates $r$, $\theta$ , where a continuously differentiable function is $f(r)$ and $a\neq 0 , b $ are constants. The exact solutions are determined for given one component of the body force, for both the cases when $f(r)$ is arbitrary and when it is not. When $f(r)$ is arbitrary, we find $a=1$ and we can construct an infinite set of streamlines and the velocity components, viscosity function, generalized energy function and temperature distribution for the cases when $R_{e}P_{r}=1$ and when $R_{e}P_{r}\neq 1$ where $R_{e}$ represents Reynolds number and $P_{r}$Prandtl number. For the case when $f(r)$ is not arbitrary we can find solutions for the cases $R_{e}P_{r}\neq a$ and $R_{e}P_{r}=a$ where $"a"$ remains arbitrary. 

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Numerical Solutions of the Viscous Flow Equations for a Class of Closed Flows

1965 ◽  
Vol 69 (658) ◽  
pp. 714-718 ◽  
Author(s):  
Ronald D. Mills
Keyword(s):  
Moving Boundary ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Navier Stokes ◽  
Steady Flows ◽  
Flow Equations ◽  
Steady Streaming ◽  
Difference Formula ◽  
Surface Passing

The Navier-Stokes equations are solved iteratively on a small digital computer for the class of flows generated within a rectangular “cavity” by a surface passing over its open end. Solutions are presented for depth/breadth ratios ƛ=0.5 (shallow), 10 (square), 20 (deep) and Reynolds number 100. Flow photographs ore obtained which largely confirm the predicted flows. The theoretical velocity profiles and pressure distributions through the centre of the vortex in the square cavity are calculated.In an appendix an improved finite difference formula is given for the vorticity generated at a moving boundary.Since Thorn began his pioneering work some thirty-five years ago the number of numerical solutions which have been obtained for the equations of incompressible viscous fluid motion remains small (see bibliographies of Thom and Apelt, Fromm). The known solutions are principally for steady streaming flows, although two methods have now been used with success for non-steady flows (Payne jets and Fromm flow past obstacles). By contrast this paper is concerned with the class of closed flows generated in a rectangular region of varying depth/breadth ratio by a surface passing over an open end. This problem has been considered for a number of reasons.

scholarly journals Critical slope for laminar transcritical shallow-water flows

2015 ◽  
Vol 783 ◽  
Author(s):  
O. Thual ◽  
L. Lacaze ◽  
M. Mouzouri ◽  
B. Boutkhamouine
Keyword(s):  
Stokes Equations ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Critical Surface ◽  
Steady Flows ◽  
Surface Slope ◽  
Critical Height ◽  
Navier Stokes Equations ◽  
Shallow Water Flows ◽  
Critical Slope

Backwater curves denote the depth profiles of steady flows in a shallow open channel. The classification of these curves for turbulent regimes is commonly used in hydraulics. When the bottom slope $I$ is increased, they can describe the transition from fluvial to torrential regimes. In the case of an infinitely wide channel, we show that laminar flows have the same critical height $h_{c}$ as that in the turbulent case. This feature is due to the existence of surface slope singularities associated to plug-like velocity profiles with vanishing boundary-layer thickness. We also provide the expression of the critical surface slope as a function of the bottom curvature at the critical location. These results validate a similarity model to approximate the asymptotic Navier–Stokes equations for small slopes $I$ with Reynolds number $Re$ such that $Re\,I$ is of order 1.

Studies on Natural Convection Induced Flow and Thermal Behavior Inside Electronic Equipment Cabinet Model

2001 ◽  
Author(s):  
Masaru Ishizuka ◽  
Guoyi Peng ◽  
Shinji Hayama
Keyword(s):  
Natural Convection ◽  
Thermal Behavior ◽  
Body Surface ◽  
Stokes Equations ◽  
The Body ◽  
Electronic Equipment ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Navier Stokes Equations ◽  
Induced Flow

Abstract In the present work, an important basic flow phenomena, the natural convection induced flow, is studied numerically. Three-dimensional Navier-Stokes equations along with the temperature equation are solved on the basis of finite difference method. Generalized coordinate system is used so that sufficient grid resolution could be achieved in the body surface boundary layer region. Differential terms with respect to time are approximated by forward differences, diffusions terms are approximated by the implicit Euler form, convection terms in the Navier-Stokes equations are approximated by the third order upwind difference scheme. The heat flux at the body surface of heater is specified. The results of calculation showed a satisfactory agreement with the measured data and led to a good understanding of the overall flow and thermal behavior inside electronic equipment cabinet model which is very difficult, if not impossible, to gather by experiment.

scholarly journals EVALUATION OF THE APPLICABILITY OF A TURBULENT WAKE INLET BOUNDARY CONDITION

2017 ◽  
Vol 16 (2) ◽  
pp. 78
Author(s):  
P. A. Soliman ◽  
A. V. de Paula ◽  
A. P. Petry ◽  
S. V. Möller
Keyword(s):  
Stokes Equations ◽  
Transport Model ◽  
Characteristic Length ◽  
Computational Cost ◽  
The Body ◽  
Navier Stokes ◽  
Computational Time ◽  
Flow Forming ◽  
Shear Stress Transport Model ◽  
Grid Convergence Index

With the objective of reducing the computational cost of the iterative processes of aerodynamic components design, tests were carried out to study under what conditions, and with what difference, only part of the calculation domain can be solved using as input information obtained from complete simulations already solved. An experimental study of an airfoil exposed to the wake interference of an upstream airfoil at a Reynolds number of 150,000 was used to verify the solutions of the Reynolds-Averaged Navier-Stokes equations solved applying the k-ω Shear Stress Transport model for turbulence closure. A Grid Convergence Index study was performed to verify if the solution of the equations for the adopted discretization leads to results within the asymptotic range. With the physical coherence of the numerical methodology verified, comparisons between the simulations with the domain comprising the two airfoils and the domain comprising only the downstream airfoil were performed. Computational time reductions in the order of 40% are observed. The differences in the aerodynamic coefficients for the two types of simulation are presented as a function of distances non-dimensionalized by the characteristic length of the body that disturbs the flow forming the wake, showing that the difference between the two methods was inversely proportional to the distance between the two bodies. Behavior that was maintained until a point where the simulation diverges, equivalent to 25% of the characteristic length of the body that generates the wake.

Studies on natural convection-induced flow and thermal behaviour inside electronic equipment cabinet model

2003 ◽  
Vol 217 (3) ◽  
pp. 323-328 ◽  
Author(s):  
M Ishizuka ◽  
Y Kitamura
Keyword(s):  
Natural Convection ◽  
Thermal Behaviour ◽  
Stokes Equations ◽  
Three Dimensional ◽  
The Body ◽  
Electronic Equipment ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Surface Boundary Layer ◽  
Induced Flow

In the present work, an important basic flow phenomenon, natural convection-induced flow, is studied numerically. Three-dimensional Navier-Stokes equations along with the energy equation are solved based on the finite difference method. A generalized coordinate system is used so that sufficient grid resolution could be achieved in the body surface boundary layer region. The results of calculation showed a satisfactory agreement with the measured data and led to a good understanding of the overall flow and thermal behaviour inside an electronic equipment cabinet model, which is very difficult, if not impossible, to gather by experiment.

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