Euler equations (fluid dynamics)

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This article is about Euler equations in classical fluid flow. For other uses, see List of topics named after Leonhard Euler.
This page assumes that classical mechanics applies; For a discussion of compressible fluid flow when velocities approach the speed of light see relativistic Euler equations.

In fluid dynamics, the Euler equations are a set of quasilinear conservation equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. The equations represent Cauchy balance equations of conservation of mass (continuity), momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and zero thermal conductivity.[1] Historically, only the continuity and momentum equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy equation – together as "the Euler equations".[2]

Like the Navier–Stokes equations, the Euler equations are usually written in one of two forms: the "conservation form" and the "lagrangian form". The conservation form emphasizes the physical interpretation of the equations as conservation laws through a control volume fixed in space. The lagrangian form emphasizes changes to the state in a frame of reference moving with the fluid.

The Euler equations can be applied to incompressible and to compressible flow– assuming that the divergence of the flow velocity field is zero, or using either as an additional constraint an appropriate equation of state respectively.

History[edit]

The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in Mémoires de l'Academie des Sciences de Berlin in 1757 (in this article Euler actually published only the general form of the continuity equation and the momentum equation;[3] the energy conservation equation would be obtained a century later). They were among the first partial differential equations to be written down. At the time Euler published his work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible fluid. An additional equation, which was later to be called the adiabatic condition, was supplied by Pierre-Simon Laplace in 1816.

During the second half of the 19th century, it was found that the equation related to the conservation of energy must at all times be kept, while the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. With the discovery of the special theory of relativity, the concepts of energy density, momentum density, and stress were unified into the concept of the stress–energy tensor, and energy and momentum were likewise unified into a single concept, the energy–momentum vector.[4]

Conservation form[edit]

The conservation form, emphasizes the physical origins of Euler equations, and expecially the contracted form is often the most convenient one for computational fluid dynamics simulations. Computationally, there are some advantages in using the conserved variables. This gives rise to a large class of numerical methods called conservative methods.[1]

Separated equations[edit]

In conservation (or Eulerian) differential form, the equations are:


\begin{align}
{\partial\rho\over\partial t}+
\nabla\cdot(\rho\bold u)=0\\[1.2ex]
{\partial(\rho{\bold u})\over\partial t}+
\nabla\cdot(\rho \bold u\otimes \bold u + p \bold I)=\bold{0}\\[1.2ex]
{\partial E\over\partial t}+
\nabla\cdot(\bold u(E+p))=0,
\end{align}

where

Note that the following identity gives the pressure gradient:

\nabla \cdot (p \bold I) = \bold I \cdot \nabla p + p \nabla \cdot \bold I = \nabla p

The equations above thus represent conservation of mass, momentum, and energy. Mass density, Flow velocity and pressure are the so-called physical variables, while mass density, momentum density and total energy density are the so-called conserved variables.[1] These equations may be expressed in subscript notation. The momentum equation (the second one) includes the divergence of a dyadic product, and may be clearer in subscript notation:


{\partial\rho\over\partial t}+
\sum_{i=1}^N
{\partial(\rho u_i)\over\partial r_i}
=0,

{\partial(\rho u_j)\over\partial t}+
\sum_{i=1}^N
{\partial(\rho u_i u_j + p \hat e_i)\over\partial r_i}
=0,

{\partial E\over\partial t}+
\sum_{i=1}^N {\partial((E+p) u_i)\over\partial r_i}
=0,

where the i and j subscripts label the N-dimensional space components: in 3D N=3 and the r and u vectors are explicitly ( x1, x2, x3 ) and ( u1, u2, u3 ). These equations may be more succinctly expressed using Einstein notation, in which matched indices imply a sum over those indices and \partial_t=\frac{\partial}{\partial t} and \partial_i=\frac{\partial}{\partial x_i}:


\partial_t \rho+\partial_i(\rho u_i)=0\,

\partial_t(\rho u_j)+\partial_i(\rho u_i u_j + e_i p)=0\,

\partial_t E+\partial_i((E+p)u_i)=0.\,

Contracted form[edit]

The Euler equations are all expressable in conservative form, so they are equivalent to a single conservation equation:

 
\frac{\partial \bold y}{\partial t}+ \nabla \cdot \bold F ={\bold 0},

or simply in Einstein notation:

 
\frac{\partial \bold y}{\partial t}+
\frac{\partial \bold f_i}{\partial r_i}={\bold 0},

where the conservation quantity y is a vector, and F is the flux matrix.


{\bold y}=\begin{pmatrix}\rho  \\  \rho \bold u  \\E\end{pmatrix};
{\bold F}=\begin{pmatrix}\rho \bold u\\\rho \bold u \otimes \bold u + p \bold I\\(E+p) \bold u\end{pmatrix};\qquad

Note that the second component containing u is by itself a vector, with length N, so y has length N+2 and F has size N(N+2). In 3D for example y has length 5, I has size 3x3 and F has size 3x5, so the explicit forms are:


{\bold y}=\begin{pmatrix}\rho \\  \rho u_1 \\ \rho u_2  \\ \rho u_3  \\E\end{pmatrix}; \quad
{\bold F}=\begin{pmatrix}\rho u_1 & \rho u_2 & \rho u_3 \\
\rho u_1^2 + p &  \rho u_1u_2 &  \rho u_1u_3
\\  \rho u_1 u_2 & \rho u_2^2 + p&  \rho u_2u_3
\\ \rho u_3 u_1 &  \rho u_3 u_2 &  \rho u_3^2 + p
\\(E+p)u_1 & (E+p)u_2 & (E+p)u_3 \end{pmatrix}.

Spatial dimensions[edit]

For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by Riemann's method of characteristics. This involves finding curves in plane of independent variables (i.e., x and t) along which partial differential equations (PDE's) degenerate into ordinary differential equations (ODE's). Numerical solutions of the Euler equations rely heavily on the method of characteristics.

Jacobian form and sound equations[edit]

Expanding the fluxes can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to the Riemann problem. From the original equations as given above in vector form, the equations are written as:

 \frac{\partial \bold y}{\partial t} + \bold A_i \frac{\partial \bold y}{\partial r_i}  = {\bold 0}.

where Ai are called the flux Jacobians defined as the matrices:

  \bold A_i (\bold y)=\frac{\partial \bold f_i (\bold y)}{\partial \bold y}.

Here, the flux Jacobians Ai are still functions of the state vector y, so this form of the Euler equations is quasilinear, just like the original equations. This Jacobian form is equivalent to the vector equation, at least in regions where the state vector y varies smoothly.

The Euler equations can be transformed into uncoupled wave equations that decribes sound in Eulerian continuum if they are expressed in characteristic variables instead of conserved variables. As an example, the one-dimensional (1-D) Euler equations in Jacobian form is considered:


\frac{\partial \bold y}{\partial t}
+ \bold A \frac{\partial \bold y}{\partial x} = {\bold 0}.

In 1D there is only one matrix A. It is diagonalizable, which means it can be decomposed with a projection matrix into a diagonal matrix:


 \mathbf A = \mathbf{P} \mathbf{D} \mathbf{P}^{-1},

\mathbf{P}=[ p_1, p_2, p_3] =\left[ 
\begin{array}{c c c}
1 & 1 & 1  \\
u-a & u & u+a \\
H-u a & \frac{1}{2} u^2 & H+u a \\
\end{array}
\right],

\mathbf{D} 
= \begin{bmatrix}
\lambda_1 & 0 & 0  \\
0 & \lambda_2 & 0 \\
0 & 0 & \lambda_3 \\
\end{bmatrix}
= \begin{bmatrix}
u-a & 0 & 0  \\
0 & u & 0 \\
0 & 0 & u+a \\
\end{bmatrix}.

Here p1, p2, p3 are the right eigenvectors of the matrix A corresponding with the eigenvalues λ1, λ2 and λ3, and the total enthalpy density is defined as:

H = E + p

Defining the characteristic variables as:

\mathbf{w}= \mathbf{P}^{-1}\mathbf{y},

Since A is constant, multiplying the original 1-D equation in flux-Jacobian form with P−1 yields:


\frac{\partial \mathbf{w_i}}{\partial t} + \lambda_j \frac{\partial \mathbf{w_i}}{\partial r_j} = \mathbf{0}

The equations have been essentially decoupled and turned into N+2 wave equations, with the eigenvalues being the wave speeds. The variables wi are called Riemann invariants or, for general hyperbolic systems, they are called characteristic variables.

Sound in ideal gas[edit]

If the equation of state is the ideal gas law, to derive the full Jacobians in matrix form, as given below:[5]

The speed of sound a is given by:


a=\sqrt{\frac{\gamma p}{\rho}} = \sqrt{(\gamma-1) h}.

where the enthalpy density h is defined (in general, not only in case of ideal gas) as:


h = e + \frac{p}{\rho} = \frac{E}{\rho} - \frac{1}{2} u^2 + \frac{p}{\rho},

Derivation[edit]

Euler equations can be obtained by linearization of some more precise equations like Navier-Stokes equations in around a local equilibrium state given by a Maxwellian y = y0, and are given by:


\frac{\partial \bold y}{\partial t}
+ \bold A_i (\bold y_0) \frac{\partial \bold y}{\partial r_i} 
= {\bold 0},

where Ai(y0), are the values of respectively Ai(y) at some reference state y = y0.

Material form[edit]

In differential material or lagrangian form, the equations are:


\begin{align}
{D\rho\over D t}+ \rho \nabla \cdot \bold u
=0\\[1.2ex]
\frac{D\bold u}{D t}+\frac {\nabla p} \rho=\bold{0}
\\[1.2ex]
{D H \over D t}  +
H \nabla \cdot \bold u - \frac{D p}{D t}=0,
\end{align}

Where the time material derivative has been used:

{D y\over D t} = {\partial y\over \partial t} + \bold u \cdot \nabla y

In fact by subtracting the velocity times the mass conservation term, the momentum equation in conservation form, can also be expressed as:


[\partial_t(\rho u_j)+\partial_i(\rho u_i u_j)+\partial_j p] - u_j[\partial_t \rho+\partial_i(\rho u_i)]=


=\rho \partial_t u_j+\rho u_i \partial_i u_j+\partial_j p=\rho D_t u_j+\partial_j p=0\,

or, in vector notation:


\rho \frac{D\bold u}{D t}+\nabla p=\bold{0}

which is a continuum mechanics form for Newton's second law of motion. Similarly, by subtracting the velocity times the above momentum conservation term, the third equation (energy conservation), can also be expressed as:


\partial_t E +\partial_i(E u_i) + p\partial_i u_i=0\,

or


\frac{\partial E}{\partial t}+\nabla\cdot(E \bold u)+p\nabla\cdot \bold u=0

that can be contracted with the material derivative as:


\frac{D E}{D t}+ (E+p) \nabla\cdot \bold u=0

or changing the variable from total energy density to total enthalpy density:


\frac{D H}{D t}+ H \nabla\cdot \bold u - \frac{D p}{D t}=0

Constraints[edit]

In N space dimensione there are thus N+2 equations and N+3 unknowns. Closing the system requires an equation of state; the most commonly used is the ideal gas law:

p =  (\gamma -1) (E - \rho \frac{u^2}{2} )

where γ is the adiabatic index.

Note the odd form for the energy equation; see Rankine–Hugoniot equation. The extra terms involving p may be interpreted as the mechanical work done on a fluid element by its neighbor fluid elements. These terms sum to zero in an incompressible fluid.

The well-known Bernoulli's equation can be derived by integrating Euler's equation along a streamline, under the assumption of constant density and a sufficiently stiff equation of state.

Steady flow in material coordinates [edit]

In the case of steady flow, it is convenient to choose the Frenet–Serret frame along a streamline as the coordinate system for describing the momentum part of the Euler equations:[6]


\frac{\mathrm{D} \boldsymbol{u}}{\mathrm{D}t} = -\frac{1}{\rho}\nabla p,

where u, p and ρ denote the flow velocity, the pressure and the density, respectively.

Let {es, en, eb } be a Frenet–Serret orthonormal basis which consists of a tangential unit vector, a normal unit vector, and a binormal unit vector to the streamline, respectively. Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-handed side of the above equation, the substantial derivative of velocity, can be described as follows:

\begin{align}
\frac{\mathrm{D} \boldsymbol{u}}{\mathrm{D}t} 
&= \boldsymbol{u}\cdot\nabla \boldsymbol{u} \\
&= u\frac{\partial}{\partial s}(u\boldsymbol{e}_s)
     &(\boldsymbol{u} = u \boldsymbol{e}_s ,~ 
      {\partial / \partial s} \equiv \boldsymbol{e}_s\cdot\nabla)\\
&= u\frac{\partial u}{\partial s}\boldsymbol{e}_s 
+ \frac{u^2}{R} \boldsymbol{e}_n &(\because~ \frac{\partial \boldsymbol{e}_s}{\partial s}=\frac{1}{R}\boldsymbol{e}_n),
\end{align}

where R is the radius of curvature of the streamline.

Therefore, the momentum part of the Euler equations for a steady flow is found to have a simple form:

\begin{cases}
\displaystyle u\frac{\partial u}{\partial s} = -\frac{1}{\rho}\frac{\partial p}{\partial s},\\
\displaystyle {u^2 \over R}                  = -\frac{1}{\rho}\frac{\partial p}{\partial n}    &({\partial / \partial n}\equiv\boldsymbol{e}_n\cdot\nabla),\\
\displaystyle 0                              = -\frac{1}{\rho}\frac{\partial p}{\partial b} &({\partial / \partial b}\equiv\boldsymbol{e}_b\cdot\nabla).
\end{cases}

For barotropic flow ( ρ=ρ(p) ), Bernoulli's equation is derived from the first equation:


\frac{\partial}{\partial s}\left(\frac{u^2}{2} + \int \frac{\mathrm{d}p}{\rho}\right) =0.

The second equation expresses that, in the case the streamline is curved, there should exist a pressure gradient normal to the streamline because the centripetal acceleration of the fluid parcel is only generated by the normal pressure gradient.

The third equation expresses that pressure is constant along the binormal axis.

Streamline curvature theorem[edit]

The "Streamline curvature theorem" states that the pressure at the upper surface of an airfoil is lower than the pressure far away and that the pressure at the lower surface is higher than the pressure far away; hence the pressure difference between the upper and lower surfaces of an airfoil generates a lift force.

Let r be the distance from the center of curvature of the streamline, then the second equation is written as follows:


\frac{\partial p}{\partial r} = \rho \frac{u^2}{r}~(>0),

where {\partial / \partial r} = -{\partial /\partial n}.

This equation states:

In a steady flow of an inviscid fluid without external forces, the center of curvature of the streamline lies in the direction of decreasing radial pressure.

Although this relationship between the pressure field and flow curvature is very useful, it doesn't have a name in the English-language scientific literature.[7] Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem". [8]

This "theorem" explains clearly why there are such low pressures in the centre of vortices,[7] which consist of concentric circles of streamlines. This also is a way to intuitively explain why airfoils generate lift forces.[7]

Shock waves[edit]

The Euler equations are quasilinear hyperbolic equations and their general solutions are waves. Much like the familiar oceanic waves, waves described by the Euler Equations 'break' and so-called shock waves are formed; this is a nonlinear effect and represents the solution becoming multi-valued. Physically this represents a breakdown of the assumptions that led to the formulation of the differential equations, and to extract further information from the equations we must go back to the more fundamental integral form. Then, weak solutions are formulated by working in 'jumps' (discontinuities) into the flow quantities – density, velocity, pressure, entropy – using the Rankine–Hugoniot shock conditions. Physical quantities are rarely discontinuous; in real flows, these discontinuities are smoothed out by viscosity and by heat transfer. (See Navier–Stokes equations)

Shock propagation is studied – among many other fields – in aerodynamics and rocket propulsion, where sufficiently fast flows occur.

See also[edit]

Notes[edit]

  1. ^ a b c see Toro, p. 24
  2. ^ Anderson, John D. (1995), Computational Fluid Dynamics, The Basics With Applications. ISBN 0-07-113210-4
  3. ^ E226 -- Principes generaux du mouvement des fluides
  4. ^ Christodoulou, Demetrios (October 2007). "The Euler Equations of Compressible Fluid Flow". Bulletin of the American Mathematical Society 44 (4): 581–602. doi:10.1090/S0273-0979-07-01181-0. Retrieved June 13, 2009. 
  5. ^ See Toro (1999)
  6. ^ James A. Fay (June 1994). Introduction to Fluid Mechanics. MIT Press. ISBN 0-262-06165-1.  see "4.5 Euler's Equation in Streamline Coordinates" pp.150-pp.152 (http://books.google.com/books?id=XGVpue4954wC&pg=150)
  7. ^ a b c Babinsky, Holger (November 2003), "How do wings work?", Physics Education 
  8. ^ 今井 功 (IMAI, Isao) (November 1973). 『流体力学(前編)』(Fluid Dynamics 1) (in Japanese). 裳華房 (Shoukabou). ISBN 4-7853-2314-0. 

Further reading[edit]

  • Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 0-521-66396-2. 
  • Thompson, Philip A. (1972). Compressible Fluid Flow. New York: McGraw-Hill. ISBN 0-07-064405-5. 
  • Toro, E.F. (1999). Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag. ISBN 3-540-65966-8.