Navier-Stokes Equations

ChessisGood · 2012-02-16T16:00:33-08:00

Navier-Stokes Equations: Greetings everyone, Today I would like to offer an interesting puzzle in the fields of mathematics, for discussion and attempted solving. A few days ago, I came across the website of the Clay Mathematics Institute. There, I found an interesting page listing seven so-called &

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ChessisGood

Feb 16, 2012

#41

The stress tensor

The derivation of the Navier-Stokes equation involves the consideration of forces acting on fluid elements, so that a quantity called the stress tensor appears naturally in the Cauchy momentum equation. Since the divergence of this tensor is taken, it is customary to write out the equation fully simplified, so that the original appearance of the stress tensor is lost.

However, the stress tensor still has some important uses, especially in formulating boundary conditions at fluid interfaces. Recalling that $\sigma_{ij} = -p I + \mathbb{T}$ , for a Newtonian fluid the stress tensor is:

$\sigma_{ij} = -p\delta_{ij}+ \mu\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) + \delta_{ij} \lambda abla \cdot \mathbf{v}.$

If the fluid is assumed to be incompressible, the tensor simplifies significantly:

$\begin{align} \sigma &= -\begin{pmatrix} p&0&0\\ 0&p&0\\ 0&0&p \end{pmatrix} + \mu \begin{pmatrix} 2 \displaystyle{\frac{\partial u}{\partial x}} & \displaystyle{\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}} &\displaystyle{ \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}} \\ \displaystyle{\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}} & 2 \displaystyle{\frac{\partial v}{\partial y}} & \displaystyle{\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}} \\ \displaystyle{\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}} & \displaystyle{\frac{\partial w}{\partial y} + \frac{\partial v}{\partial z}} & 2\displaystyle{\frac{\partial w}{\partial z}} \end{pmatrix} \\ &= -p I + \mu ( abla \mathbf{v} + ( abla \mathbf{v})^T) = -p I + 2 \mu e\\ \end{align}$

$e$ is the strain rate tensor, by definition:

$e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$

ChessisGood

Feb 16, 2012

#42

Notes

^ ^a ^b Lebedev, Leonid P. (2003). Tensor Analysis. World Scientific. ISBN 9812383603.
^ Batchelor 2000, pp. 141.
^ Batchelor 2000, pp. 144.
^ Eric W. Weisstein. "Vector Derivative". MathWorld. Retrieved 2008-06-07.

ChessisGood

Feb 16, 2012

#43

References

Batchelor, G.K. (2000). An Introduction to Fluid Dynamics. New York: Cambridge University Press. ISBN 978-0-521-66396-0
White, Frank M. (2006). Viscous Fluid Flow (3rd ed.). New York, NY: McGraw Hill. ISBN 0-07-240231-8
Surface Tension Module, by John W. M. Bush, at MIT OCW.

ChessisGood

Feb 16, 2012

#44

Very simple:

http://mathworld.wolfram.com/Navier-StokesEquations.html

avier-Stokes Equations

The equation of incompressible fluid flow,

where is the kinematic viscosity, is the velocity of the fluid parcel, is the pressure, and is the fluid density.

The Navier-Stokes equations appear in Big Weld's office in the 2005 animated film Robots.

REFERENCES:

Landau, L. D. and Lifschitz, E. M. Fluid Mechanics, 2nd ed. Oxford, England: Pergamon Press, p. 15, 1982.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 139, 1997.

ChessisGood

Feb 16, 2012

#45

Nice information on the NASA website:

http://www.grc.nasa.gov/WWW/k-12/airplane/nseqs.html

ChessisGood

Feb 16, 2012

#46

On this slide we show the three-dimensional unsteady form of the Navier-Stokes Equations. These equations describe how the velocity, pressure, temperature, and density of a moving fluid are related. The equations were derived independently by G.G. Stokes, in England, and M. Navier, in France, in the early 1800's. The equations are extensions of the Euler Equations and include the effects of viscosity on the flow. These equations are very complex, yet undergraduate engineering students are taught how to derive them in a process very similar to the derivation that we present on theconservation of momentum web page.

The equations are a set of coupled differential equations and could, in theory, be solved for a given flow problem by using methods from calculus. But, in practice, these equations are too difficult to solve analytically. In the past, engineers made further approximations and simplifications to the equation set until they had a group of equations that they could solve. Recently, high speed computers have been used to solve approximations to the equations using a variety of techniques like finite difference, finite volume, finite element, and spectral methods. This area of study is called Computational Fluid Dynamics or CFD.

The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass, three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation. There are four independent variables in the problem, the x, y, and z spatial coordinates of some domain, and the time t. There are six dependent variables; the pressure p, density r, and temperature T (which is contained in the energy equation through the total energy Et) and three components of the velocity vector; the u component is in the x direction, the v component is in the y direction, and the w component is in the z direction, All of the dependent variables are functions of all four independent variables. The differential equations are therefore partial differential equations and not the ordinary differential equations that you study in a beginning calculus class.

ChessisGood

Feb 16, 2012

#47

You will notice that the differential symbol is different than the usual "d /dt" or "d /dx" that you see for ordinary differential equations. The symbol "" is is used to indicate partial derivatives. The symbol indicates that we are to hold all of the independent variables fixed, except the variable next to symbol, when computing a derivative. The set of equations are:

Continuity: r/t + (r * u)/x + (r * v)/y + (r * w)/z = 0

X - Momentum: (r * u)/t + (r * u^2)/x + (r * u * v)/y + (r * u * w)/z = - p/x

+ 1/Re * { tauxx/x + tauxy/y + tauxz/z}

Y - Momentum: (r * v)/t + (r * u * v)/x + (r * v^2)/y + (r * v * w)/z = - p/y

+ 1/Re * { tauxy/x + tauyy/y + tauyz/z}

Z - Momentum: (r * w)/t + (r * u * w)/x + (r * v * w)/y + (r * w^2)/z = - p/z

+ 1/Re * { tauxz/x + tauyz/y + tauzz/z}

Energy: Et/t + (u * Et)/x + (v * Et)/y + (w * Et)/z = - (r * u)/x - (r * v)/y - (r * w)/z

- 1/(Re*Pr) * { qx/x + qy/y + qz/z}

+ 1/Re * {(u * tauxx + v * tauxy + w * tauxz)/x + (u * tauxy + v * tauyy + w * tauxz)/y + (u * tauxz + v * tauyz + w * tauzz)/z}

where Re is the Reynolds number which is a similarity parameter that is the ratio of the scaling of the inertia of the flow to the viscous forces in the flow. The q variables are the heat flux components and Pr is the Prandtl number which is a similarity parameter that is the ratio of the viscous stresses to the thermal stresses. The tau variables are components of the stress tensor. A tensor is generated when you multiply two vectors in a certain way. Our velocity vector has three components; the stress tensor has nine components. Each component of the stress tensor is itself a second derivative of the velocity components.

The terms on the left hand side of the momentum equations are called the convection terms of the equations. Convectionis a physical process that occurs in a flow of gas in which some property is transported by the ordered motion of the flow. The terms on the right hand side of the momentum equations that are multiplied by the inverse Reynolds number are called the diffusion terms. Diffusion is a physical process that occurs in a flow of gas in which some property is transported by the random motion of the molecules of the gas. Diffusion is related to the stress tensor and to the viscosity of the gas. Turbulence, and the generation of boundary layers, are the result of diffusion in the flow. The Euler equations contain only the convection terms of the Navier-Stokes equations and can not, therefore, model boundary layers. There is a special simplification of the Navier-Stokes equations that describe boundary layer flows.

Notice that all of the dependent variables appear in each equation. To solve a flow problem, you have to solve all five equations simultaneously; that is why we call this a coupled system of equations. There are actually some other equation that are required to solve this system. We only show five equations for six unknowns. An equation of state relates the pressure, temperature, and density of the gas. And we need to specify all of the terms of the stress tensor. In CFD the stress tensor terms are often approximated by a turbulence model.

ChessisGood

Feb 16, 2012

#48

Very picture oriented:

http://www.cfd-online.com/Wiki/Navier-Stokes_equations

The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Newton's Law of Motion to a fluid element and is also called the momentum equation. It is supplemented by the mass conservation equation, also called continuity equation and the energy equation. Usually, the term Navier-Stokes equations is used to refer to all of these equations.

ChessisGood

Feb 16, 2012

#49

Equations

The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:

$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0$	(1)
$\frac{\partial}{\partial t}\left( \rho u_i \right) + \frac{\partial}{\partial x_j} \left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0, \quad i=1,2,3$	(2)
$\frac{\partial}{\partial t}\left( \rho e_0 \right) + \frac{\partial}{\partial x_j} \left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0$

ChessisGood

Feb 16, 2012

#50

For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:

$\tau_{ij} = 2 \mu S_{ij}^*$

ChessisGood

Feb 16, 2012

#51

Where the trace-less viscous strain-rate is defined by:

$S_{ij}^* \equiv \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$

ChessisGood

Feb 16, 2012

#52

The heat-flux, $q_j$ , is given by Fourier's law:

$q_j = -\lambda \frac{\partial T}{\partial x_j} \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}$

ChessisGood

Feb 16, 2012

#53

Where the laminar Prandtl number $Pr$ is defined by:

$Pr \equiv \frac{C_p \mu}{\lambda}$

ChessisGood

Feb 16, 2012

#54

To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:

$\gamma \equiv \frac{C_p}{C_v} ~~,~~ p = \rho R T ~~,~~ e = C_v T ~~,~~ C_p - C_v = R$

(8)

Where $\gamma$ , $C_p$ , $C_v$ and $R$ are constant.

ChessisGood

Feb 16, 2012

#55

The total energy $e_0$ is defined by:

$e_0 \equiv e + \frac{u_k u_k}{2}$

ChessisGood

Feb 16, 2012

#56

Note that the corresponding expression (15) for Favre averaged turbulent flows contains an extra term related to the turbulent energy.

Equations (1)-(9), supplemented with gas data for $\gamma$ , $Pr$ , $\mu$ and perhaps $R$ , form a closed set of partial differential equations, and need only be complemented with boundary conditions.

ChessisGood

Feb 16, 2012

#57

Derivation

Derivation of continuity equation

A fundamental laws of Newtonian mechanics states the conservation of mass in an arbitrary material control volume varying in time $V_{m}$ .

ChessisGood

Feb 16, 2012

#58

A material volume contains the same portions of a fluid at all times. It may be defined by a closed bounding surface $S_{m}$ eneveloping a portion of a fluid at certain time. Fluid elements cannot enter or leave this control volume. The movement of every point on the surface $S_{m}$ is defined by the local velocity $\boldsymbol{u}$ . So one can define:

$0= \frac {dM}{dt}= \frac {d}{dt} \int_{V_{m}} \rho \, dV.$

Applying the Reynolds transport theorem and divergence theorem one obtains:

$0= \frac {dM}{dt}= \int_{V_{m}} \frac {\partial \rho}{\partial t} dV + \int_{S_m} \rho \boldsymbol{u}\cdot\boldsymbol{n} \, dS = \int_{V_{m}} \left( \frac{\partial \rho}{\partial t} + \boldsymbol{ abla} \cdot \left( \rho \boldsymbol{u} \right) \right) dV$

Since this relation is valid for an arbitrary volume $V_{m}$ , the integrand must be zero. Note that now it can easily be assumed that the volume is a fixed control volume (where fluid particles can freely enter and leave the volume) by taking account of mass fluxes through the surface $S_{m}$ .

Thus

$\frac{\partial \rho}{\partial t} + \boldsymbol{ abla} \cdot \left( \rho \boldsymbol{u} \right)$

(1)

at all points of the fluid. For an incompressible fluid the change rate of density is zero. One can simplify (1) to:

$\boldsymbol{ abla} \cdot \boldsymbol{u} = 0.$

ChessisGood

Feb 16, 2012

#59

Derivation of momentum equations

Expansion, rotation and deformation of a fluid parsel

ChessisGood

Feb 16, 2012

#60

forces and stresses

$\textbf{V} = \textbf{i}u + \textbf{j}v + \textbf{k}w$

(2)

$\rho \frac{D \textbf{V}}{Dt} = \rho \textbf{F} + \textbf{P}$

(3)

where

$\textbf{F}$ - mass force per volume unit

$\textbf{P}$ - surface force per volume unit

$\textbf{F} = \textbf{i}F_{x} + \textbf{j}F_{y} + \textbf{k}F_{z}$