Euler equations (fluid dynamics)

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 equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass (continuity), momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. 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".

Like the Navier-Stokes equations, the Euler equations are usually written in one of two forms: the "conservation form" and the "non-conservation form". The conservation form emphasizes the physical interpretation of the equations as conservation laws through a control volume fixed in space. The non-conservation form emphasises changes to the state of a control volume as it moves with the fluid.

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

History

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. 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, thus it 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.

Conservation and component form

In differential form, the equations are:

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

where

• ρ is the fluid mass density,
• u is the fluid velocity vector, with components u, v, and w,
• E = ρ e + ½ ρ ( u² + v² + w² ) is the total energy per unit volume, with e being the internal energy per unit mass for the fluid,
• p is the pressure,
• \(\otimes\) denotes the tensor product, and
• \(\bold{0}\) being the zero vector.

These equations may be expressed in subscript notation. The second equation includes the divergence of a dyadic product, and may be clearer in subscript notation:

\[ {\partial\rho\over\partial t}+ \sum_{i=1}^3 {\partial(\rho u_i)\over\partial x_i} =0, \]

\[ {\partial(\rho u_j)\over\partial t}+ \sum_{i=1}^3 {\partial(\rho u_i u_j)\over\partial x_i}+ {\partial p\over\partial x_j} =0, \]

\[ {\partial E\over\partial t}+ \sum_{i=1}^3 {\partial((E+p) u_i)\over\partial x_i} =0, \]

where the i and j subscripts label the three Cartesian components: ( x₁ , x₂ , x₃ ) = ( x , y , z ) and ( u₁ , u₂ , u₃ ) = ( u , v , w ). These equations may be more succinctly expressed using abstract index notation in which matched indices imply a sum over those indices and \(\partial_t=\frac{\partial}{\partial t}\) and \(\partial_i=\frac{\partial}{\partial x_i}\):

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

\[ \partial_t(\rho u_j)+\partial_i(\rho u_i u_j)+\partial_j p=0\, \]

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

Note that the above equations are expressed in conservation form, as this format emphasizes their physical origins (and is often the most convenient form for computational fluid dynamics simulations). By subtracting the velocity times the mass conservation term, the second equation (momentum conservation), can also be expressed as:

\[ [\partial_t(\rho u_j)+\partial_i(\rho u_i u_j)+\partial_j p] - u_j[\partial_t \rho+\partial_i(\rho u_i)]= \rho \partial_t u_j+\rho u_i \partial_i u_j+\partial_j p=0\, \]

or, in vector notation:

\[ \rho\left( \frac{\partial}{\partial t}+{\bold u}\cdot\nabla \right){\bold u}+\nabla p=\bold{0} \]

but this form for the momentum conservation equation obscures the direct connection between the Euler equations and 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:

\[ \partial_t e+\partial_i(eu_i) + p\partial_i u_i=0\, \]

or

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

Conservation and vector form

In vector and conservation form, the Euler equations become:

\[ \frac{\partial \bold m}{\partial t}+ \frac{\partial \bold f_x}{\partial x}+ \frac{\partial \bold f_y}{\partial y}+ \frac{\partial \bold f_z}{\partial z}={\bold 0}, \]

where

\[ {\bold m}=\begin{pmatrix}\rho \\ \rho u \\ \rho v \\ \rho w \\E\end{pmatrix}; \]

\[ {\bold f_x}=\begin{pmatrix}\rho u\\p+\rho u^2\\ \rho uv \\ \rho uw\\u(E+p)\end{pmatrix};\qquad {\bold f_y}=\begin{pmatrix}\rho v\\ \rho uv \\p+\rho v^2\\ \rho vw \\v(E+p)\end{pmatrix};\qquad {\bold f_z}=\begin{pmatrix}\rho w\\ \rho uw \\ \rho vw \\p+\rho w^2\\w(E+p)\end{pmatrix}. \]

This form makes it clear that f_x, f_y and f_z are fluxes.

The equations above thus represent conservation of mass, three components of momentum, and energy. There are thus five equations and six unknowns. Closing the system requires an equation of state; the most commonly used is the ideal gas law (i.e. p = ρ (γ−1) e, where ρ is the density, γ is the adiabatic index, and e the internal energy).

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.

Non-conservation form with flux Jacobians

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 and conservation form, the equations are written in a non-conservation form as:

\[ \frac{\partial \bold m}{\partial t} + \bold A_x \frac{\partial \bold m}{\partial x} + \bold A_y \frac{\partial \bold m}{\partial y} + \bold A_z \frac{\partial \bold m}{\partial z} = {\bold 0}. \]

where A_x, A_y and A_z are called the flux Jacobians, which are matrices equal to:

\[ \bold A_x=\frac{\partial \bold f_x(\bold s)}{\partial \bold s}, \qquad \bold A_y=\frac{\partial \bold f_y(\bold s)}{\partial \bold s} \qquad \text{and} \qquad \bold A_z=\frac{\partial \bold f_z(\bold s)}{\partial \bold s}. \]

Here, the flux Jacobians A_x, A_y and A_z are still functions of the state vector m, so this form of the Euler equations is nonlinear, just like the original equations. This non-conservation form is equivalent to the original Euler equations in conservation form, at least in regions where the state vector m varies smoothly.

Flux Jacobians for an ideal gas

The ideal gas law is used as the equation of state, to derive the full Jacobians in matrix form, as given below:

Flux Jacobians in matrix form for an ideal gas

The x-direction flux Jacobian: \[ \bold A_x= \left[ \begin{array}{c c c c c} 0 & 1 & 0 & 0 & 0 \\ \hat{\gamma}H-u^2-a^2 & (3-\gamma)u & -\hat{\gamma}v & -\hat{\gamma}w & \hat{\gamma} \\ -uv & v & u & 0 & 0 \\ -uw & w & 0 & u & 0 \\ u[(\gamma-2)H-a^2] & H-\hat{\gamma}u^2 & -\hat{\gamma}uv & -\hat{\gamma}uw & \gamma u \end{array} \right]. \] The y-direction flux Jacobian: \[ \bold A_y= \left[ \begin{array}{c c c c c} 0 & 0 & 1 & 0 & 0 \\ -vu & v & u & 0 & 0 \\ \hat{\gamma}H-v^2-a^2 & -\hat{\gamma}u & (3-\gamma)v & -\hat{\gamma}w & \hat{\gamma} \\ -vw & 0 & w & v & 0 \\ v[(\gamma-2)H-a^2] & -\hat{\gamma}uv & H-\hat{\gamma}v^2 & -\hat{\gamma}vw & \gamma v \end{array} \right]. \] The z-direction flux Jacobian: \[ \bold A_z= \left[ \begin{array}{c c c c c} 0 & 0 & 0 & 1 & 0 \\ -uw & w & 0 & u & 0 \\ -vw & 0 & w & v & 0 \\ \hat{\gamma}H-w^2-a^2 & -\hat{\gamma}u & -\hat{\gamma}v & (3-\gamma)w& \hat{\gamma} \\ w[(\gamma-2)H-a^2] & -\hat{\gamma}uw & -\hat{\gamma}vw & H-\hat{\gamma}w^2 & \gamma w \end{array} \right]. \] Where \(\hat{\gamma}=\gamma-1\).

The total enthalpy H is given by:

\[ H = \frac{E}{\rho} + \frac{p}{\rho}, \]

and the speed of sound a is given as:

\[ a=\sqrt{\frac{\gamma p}{\rho}} = \sqrt{(\gamma-1)\left[H-\frac{1}{2}\left(u^2+v^2+w^2\right)\right]}. \]

Linearized form

The linearized Euler equations are obtained by linearization of the Euler equations in non-conservation form with flux Jacobians, around a state m = m₀, and are given by:

\[ \frac{\partial \bold m}{\partial t} + \bold A_{x,0} \frac{\partial \bold m}{\partial x} + \bold A_{y,0} \frac{\partial \bold m}{\partial y} + \bold A_{z,0} \frac{\partial \bold m}{\partial z} = {\bold 0}, \]

where A_x,0, A_y,0 and A_z,0 are the values of respectively A_x, A_y and A_z at some reference state m = m₀.

Uncoupled wave equations for the linearized one-dimensional case

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

\[ \frac{\partial \bold m}{\partial t} + \bold A_{x,0} \frac{\partial \bold m}{\partial x} = {\bold 0}. \]

The matrix A_x,0 is diagonalizable, which means it can be decomposed into:

\[ \mathbf{A}_{x,0} = \mathbf{P} \mathbf{\Lambda} \mathbf{P}^{-1}, \]

\[ \mathbf{P}= \left[\bold r_1, \bold r_2, \bold r_3\right] =\left[ \begin{array}{c c c} 1 & 1 & 1 \\ u-a & u & u+a \\ H-u a & \frac{1}{2} u^2 & H+u a \\ \end{array} \right], \]

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

Here r₁, r₂, r₃ are the right eigenvectors of the matrix A_x,0 corresponding with the eigenvalues λ₁, λ₂ and λ₃.

Defining the characteristic variables as:

\[\mathbf{w}= \mathbf{P}^{-1}\mathbf{m},\]

Since A_x,0 is constant, multiplying the original 1-D equation in flux-Jacobian form with P⁻¹ yields:

\[ \frac{\partial \mathbf{w}}{\partial t} + \mathbf{\Lambda} \frac{\partial \mathbf{w}}{\partial x} = \mathbf{0} \]

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

Shock waves

The Euler equations are nonlinear 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. (See Navier–Stokes equations)

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

The equations in one spatial dimension

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.

Steady flow in streamline coordinates

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: \[ {\mathrm{D} \boldsymbol{v} \over \mathrm{D}t} = -{1\over\rho}\nabla p, \] where \(\boldsymbol{v}\), \(p\) and \(\rho\) denote the velocity, the pressure and the density, respectively.

Let \(\{\boldsymbol{e}_s, \boldsymbol{e}_n, \boldsymbol{e}_b\}\) 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: \[\begin{align} {\mathrm{D} \boldsymbol{v} \over \mathrm{D}t} &= \boldsymbol{v}\cdot\nabla \boldsymbol{v} \\ &= v{\partial \over \partial s}(v\boldsymbol{e}_s) &(\boldsymbol{v} = v \boldsymbol{e}_s ,~ {\partial / \partial s} \equiv \boldsymbol{e}_s\cdot\nabla)\\ &= v{\partial v \over \partial s}\boldsymbol{e}_s + {v^2 \over R} \boldsymbol{e}_n &(\because~ {\partial \boldsymbol{e}_s \over \partial s}={1\over R}\boldsymbol{e}_n), \end{align}\] 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: \[\begin{cases} \displaystyle v{\partial v \over \partial s} = -{1 \over \rho}{\partial p \over \partial s},\\ \displaystyle {v^2 \over R} = -{1 \over \rho}{\partial p \over \partial n} &({\partial / \partial n}\equiv\boldsymbol{e}_n\cdot\nabla),\\ \displaystyle 0 = -{1 \over \rho}{\partial p \over \partial b} &({\partial / \partial b}\equiv\boldsymbol{e}_b\cdot\nabla). \end{cases} \]

For barotropic flow ( \(\rho=\rho(p)\) ), Bernoulli's equation is derived from the first equation: \[ {\partial \over \partial s}\left({v^2\over 2} + \int {\mathrm{d}p \over \rho}\right) =0. \] 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

File:Streamlines around a NACA 0012.svg

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: \[ {\partial p \over \partial r} = \rho {v^2 \over r}~(>0), \] 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. Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem".

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

See also

• Madelung equations

Further reading

• bg:Уравнения на Ойлер