- Euler equations (fluid dynamics)
In
fluid dynamics , the Euler equations governinviscid flow . They correspond to theNavier-Stokes equations with zeroviscosity andheat conduction terms. They are usually written in the conservation form shown below to emphasize that they directly representconservation of mass , momentum, and energy. The equations are named afterLeonhard Euler .The Euler equations can be applied to compressible as well as to
incompressible flow — using either an appropriateequation of state or that thedivergence of theflow velocity field is zero, repectively.This page assumes that
classical mechanics applies; seerelativistic Euler equations for a discussion of compressible fluid flow when velocities approach thespeed of light .Euler equations in conservation and component form
In differential form, the equations are:
:egin{align}&{partial hooverpartial t}+ ablacdot( hoold u)=0\ [1.2ex] &{partial ho{old u}overpartial t}+ ablacdot(old uotimes( ho old old u))+ abla p=0\ [1.2ex] &{partial Eoverpartial t}+ ablacdot(old u(E+p))=0,end{align}
where
*"ρ" is the fluidmass density ,
*"u" is the fluidvelocity vector, with components "u", "v", and "w",
*"E = ρ e + ½ ρ ( u2 + v2 + w2 )" is the totalenergy per unitvolume , with "e" is theinternal energy per unit mass for the fluid, and
*"p" is thepressure .The second equation includes thedivergence of adyadic product , and may be clearer in subscript notation; for each "j" from 1 to 3 one has::partial( ho u_j)overpartial t}+sum_{i=1}^3{partial( ho u_i u_j)overpartial x_i}+{partial poverpartial x_j}=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 )".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). The second equation, which representsmomentum conservation, can also be expressed in non-conservation form as::holeft(frac{partial}{partial t}+{old u}cdot abla ight){old u}+ abla p=0
but this form obscures the direct connection between the Euler equations and
Newton's second law of motion .Euler equations in conservation and vector form
In vector and conservation form, the Euler equations become:
:frac{partial old m}{partial t}+frac{partial old f_x}{partial x}+frac{partial old f_y}{partial y}+frac{partial old f_z}{partial z}=0,
where
:old m}=egin{pmatrix} ho \ ho u \ ho v \ ho w \Eend{pmatrix}qquad{old f_x}=egin{pmatrix} ho u\p+ ho u^2\ ho uv \ ho uw\u(E+p)end{pmatrix}qquad{old f_y}=egin{pmatrix} ho v\ ho uv \p+ ho v^2\ ho vw \v(E+p)end{pmatrix}qquad{old f_z}=egin{pmatrix} ho w\ ho uw \ ho vw \p+ ho w^2\w(E+p)end{pmatrix}.
This form makes it clear that "fx", "fy" and "fz" are
flux es.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 anequation of state ; the most commonly used is theideal gas law (i.e. "p = ρ (γ-1) e", where "ρ" is the density, "γ" is theadiabatic 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.Euler equations in non-conservation form with flux Jacobians
Expanding the
flux es can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to theRiemann 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 old m}{partial t}+ old A_x frac{partial old m}{partial x} + old A_y frac{partial old m}{partial y} + old A_z frac{partial old m}{partial z} = 0.
where A"x", A"y" and A"z" are called the flux
Jacobian s, which are matrices equal to::old A_x=frac{partial old f_x(old s)}{partial old s}, qquad old A_y=frac{partial old f_y(old s)}{partial old s} qquad ext{and} qquad old A_z=frac{partial old f_z(old s)}{partial old 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 theequation of state , to derive the full Jacobians in matrix form, as given below [See Toro (1999)] ::
The total
enthalpy "H" is given by::H =E+frac{p}{ ho},
and the
speed of sound "a" is given as::a=sqrt{frac{gamma p}{ ho = sqrt{(gamma-1)left [H-frac{1}{2}left(u^2+v^2+w^2 ight) ight] }.
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" = "m"0, and are given by:
:frac{partial old m}{partial t}+ old A_{x,0} frac{partial old m}{partial x} + old A_{y,0} frac{partial old m}{partial y} + old A_{z,0} frac{partial old m}{partial z} = 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" = "m"0.
Transformation to uncoupled wave equations for the 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 old m}{partial t}+ old A_{x,0} frac{partial old m}{partial x} =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 [old r_1, old r_2, old r_3 ight] =left [ egin{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} ight] ,
:mathbf{Lambda} = egin{bmatrix}lambda_1 & 0 & 0 \0 & lambda_2 & 0 \0 & 0 & lambda_3 \end{bmatrix}= egin{bmatrix}u-a & 0 & 0 \0 & u & 0 \0 & 0 & u+a \end{bmatrix}.
Here "r1", "r2", "r3" are the
right eigenvector s of the matrix A"x,0" corresponding with theeigenvalue s "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} = 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".
hock waves
The Euler equations are
nonlinear hyperbolic equations and their general solutions arewaves . Much like the familiar oceanic waves, waves described by the Euler Equations 'break' and so-calledshock 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 byviscosity . (SeeNavier-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 whichpartial differential equation s (PDE's) degenerate intoordinary differential equation s (ODE's). Numerical solutions of the Euler equations rely heavily on the method of characteristics.References
*cite book | first=G. K. | last=Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0521663962
*cite book | first=Philip A. | last=Thompson| year=1972 | title=Compressible Fluid Flow | publisher=McGraw-Hill | location=New York | isbn=0070644055
*cite book | first=E.F. | last=Toro | title=Riemann Solvers and Numerical Methods for Fluid Dynamics | publisher=Springer-Verlag | year=1999 | isbn=3-540-65966-8Notes
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