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{{Short description|Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow}}{{About|Euler equations in classical fluid flow||List of topics named after Leonhard Euler}}{{Hatnote|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.}}(File:Flow around a wing.gif|thumb|Flow around a wing. This incompressible flow satisfies the Euler equations.)In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity.{{sfn|Toro|1999|p= 24}}The Euler equations can be applied to incompressible and compressible flows. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is a solenoidal field. The compressible Euler equations consist of equations for conservation of mass, balance of momentum, and balance of energy, together with a suitable constitutive equation for the specific energy density of the fluid. Historically, only the equations of conservation of mass and balance of momentum were derived by Euler. However, fluid dynamics literature often refers to the full set of the compressible Euler equations – including the energy equation – as "the compressible Euler equations".{{sfn|Anderson|1995|p=}}The mathematical characters of the incompressible and compressible Euler equations are rather different. For constant fluid density, the incompressible equations can be written as a quasilinear advection equation for the fluid velocity together with an elliptic Poisson's equation for the pressure. On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations.The Euler equations can be formulated in a "convective form" (also called the "Lagrangian form") or a "conservation form" (also called the "Eulerian form"). The convective form emphasizes changes to the state in a frame of reference moving with the fluid. The conservation form emphasizes the mathematical interpretation of the equations as conservation equations for a control volume fixed in space (which is usefulfrom a numerical point of view).

History

The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in Mémoires de l'Académie des Sciences de Berlin in 1757{{sfn|Euler|1757}} (although Euler had previously presented his work to the Berlin Academy in 1752).{{sfn|Christodoulou|2007|p=}}The Euler equations were among the first partial differential equations to be written down, after the wave equation. In Euler's original work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible flow. An additional equation, which was 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 balance of energy must at all times be kept for compressible flows, and 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.{{sfn|Christodoulou|2007|p=}}

Incompressible Euler equations with constant and uniform density

In convective form (i.e., the form with the convective operator made explicit in the momentum equation), the incompressible Euler equations in case of density constant in time and uniform in space are:{{sfn|Hunter|2006|p=}}{{Equation box 1|indent=:|title=Incompressible Euler equations with constant and uniform density(convective or Lagrangian form)|equation=left{begin{align} end{align}right.|cellpadding|border|border colour = #0073CF|background colour=#F5FFFA}}where:

• mathbf u is the flow velocity vector, with components in an N-dimensional space u_1, u_2, dots, u_N,
• frac{DboldsymbolPhi}{Dt} = frac{partialboldsymbolPhi}{partial t} + mathbf v cdot nabla boldsymbolPhi, for a generic function (or field) boldsymbolPhi denotes its material derivative in time with respect to the advective field mathbf v and
• nabla w is the gradient of the specific (with the sense of per unit mass) thermodynamic work, the internal source term, and
• nabla cdot mathbf u is the flow velocity divergence.
• mathbf{g} represents body accelerations (per unit mass) acting on the continuum, for example gravity, inertial accelerations, electric field acceleration, and so on.

The first equation is the Euler momentum equation with uniform density (for this equation it could also not be constant in time). By expanding the material derivative, the equations become:left{begin{align} end{align}right.In fact for a flow with uniform density rho_0 the following identity holds:nabla w equiv nabla left(frac p {rho_0} right) = frac 1 {rho_0} nabla p where p is the mechanic pressure. The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field (the order of the equations is not causal, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following). Notably, the continuity equation would be required also in this incompressible case as an additional third equation in case of density varying in time or varying in space. For example, with density uniform but varying in time, the continuity equation to be added to the above set would correspond to:frac{partial rho}{partial t} = 0 So the case of constant and uniform density is the only one not requiring the continuity equation as additional equation regardless of the presence or absence of the incompressible constraint. In fact, the case of incompressible Euler equations with constant and uniform density discussed here is a toy model featuring only two simplified equations, so it is ideal for didactical purposes even if with limited physical relevance.The equations above thus represent respectively conservation of mass (1 scalar equation) and momentum (1 vector equation containing N scalar components, where N is the physical dimension of the space of interest). Flow velocity and pressure are the so-called physical variables.{{sfn|Toro|1999|p= 24}}In a coordinate system given by left(x_1, dots, x_Nright) the velocity and external force vectors mathbf u and mathbf g have components (u_1,dots, u_N) and left(g_1, dots, g_Nright), respectively. Then the equations may be expressed in subscript notation as: end{align}right.where the i and j subscripts label the N-dimensional space components, and delta_{ij} is the Kroenecker delta. The use of Einstein notation (where the sum is implied by repeated indices instead of sigma notation) is also frequent.

Properties

Although Euler first presented these equations in 1755, many fundamental questions or concepts about them remain unanswered.In three space dimensions, in certain simplified scenarios, the Euler equations produce singularities.JOURNAL, Elgindi, Tarek M., 2021-11-01, Finite-time singularity formation for $C^{1,alpha}$ solutions to the incompressible Euler equations on $mathbb{R}^3$,weblink Annals of Mathematics, 194, 1904.04795, 3, 10.4007/annals.2021.194.3.2, 0003-486X, Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy:{partial overpartial t} left(frac{1}{2} u^2 right) + nabla cdot left(u^2 mathbf u + w mathbf uright) = 0 In the one-dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid Burgers' equation:{partial u overpartial t}+ u {partial u overpartial x} = 0This model equation gives many insights into Euler equations.

Nondimensionalisation

{{See also|Cauchy momentum equation#Nondimensionalisation}}{{Unreferenced section|date=April 2021}}In order to make the equations dimensionless, a characteristic length r_0, and a characteristic velocity u_0, need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:begin{align} end{align}and of the field unit vector:hat{mathbf g}equiv frac {mathbf g} g, Substitution of these inversed relations in Euler equations, defining the Froude number, yields (omitting the * at apix):{hide}Equation box 1|indent=:|title=Incompressible Euler equations with constant and uniform density(nondimensional form)|equation=left{begin{align} end{align}right.|cellpadding|border|border colour = #0073CF|background colour=#F5FFFA}}Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory.

Conservation form

{{See also|Conservation equation|}}The conservation form emphasizes the mathematical properties of Euler equations, and especially 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 methodscalled conservative methods.{{sfn|Toro|1999|p= 24}}The free Euler equations are conservative, in the sense they are equivalent to a conservation equation:frac{partial mathbf y}{partial t}+ nabla cdot mathbf F ={mathbf 0},or simply in Einstein notation:frac{partial y_j}{partial t}+frac{partial f_{ij}}{partial r_i}= 0_i,where the conservation quantity mathbf y in this case is a vector, and mathbf F is a flux matrix. This can be simply proved.{{hidden|Demonstration of the conservation form|First, the following identities hold:nabla cdot (w mathbf I) = mathbf I cdot nabla w + w nabla cdot mathbf I = nabla w mathbf u cdot nabla cdot mathbf u = nabla cdot (mathbf u otimes mathbf u)where otimes denotes the outer product. The same identities expressed in Einstein notation are:partial_ileft(w delta_{ij}right) = delta_{ij} partial_i w + w partial_i delta_{ij} = delta_{ij} partial_i w = partial_j wu_j partial_i u_i = partial_i left(u_i u_jright)where {{mvar|I}} is the identity matrix with dimension {{mvar|N}} and {{mvar|δij}} its general element, the Kroenecker delta.Thanks to these vector identities, the incompressible Euler equations with constant and uniform density and without external field can be put in the so-called conservation (or Eulerian) differential form, with vector notation:left{begin{align} end{align}right.or with Einstein notation:left{begin{align} end{align}right.Then incompressible Euler equations with uniform density have conservation variables:mathbf y = begin{pmatrix}mathbf u 0 end{pmatrix}; qquadmathbf F = begin{pmatrix}mathbf u otimes mathbf u + w mathbf I mathbf u end{pmatrix}.Note that in the second component u is by itself a vector, with length N, so y has length N+1 and F has size N(N+1). In 3D for example y has length 4, I has size 3×3 and F has size 4×3, so the explicit forms are:{mathbf y}=begin{pmatrix} u_1 u_2 u_3 end{pmatrix}; quad{mathbf F}=begin{pmatrix}u_1^2 + w & u_1u_2 & u_1u_3 u_2 u_1 & u_2^2 + w & u_2 u_3 u_3 u_1 & u_3 u_2 & u_3^2 + w u_1 & u_2 & u_3 end{pmatrix}.|style = border: 1px solid lightgray; width: 90%;|headerstyle = text-align:left;}}At last Euler equations can be recast into the particular equation:{{Equation box 1|indent=:|title=Incompressible Euler equation(s) with constant and uniform density(conservation or Eulerian form)|equation=frac {partial}{partial t}begin{pmatrix} mathbf u 0 end{pmatrix} + nabla cdot begin{pmatrix}mathbf u otimes mathbf u + w mathbf I mathbf u end{pmatrix} = begin{pmatrix}mathbf g 0end{pmatrix}|cellpadding|border|border colour = #0073CF|background colour=#F5FFFA}}

Spatial dimensions

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 (PDEs) degenerate into ordinary differential equations (ODEs). Numerical solutions of the Euler equations rely heavily on the method of characteristics.

Incompressible Euler equations

In convective form the incompressible Euler equations in case of density variable in space are:{{sfn|Hunter|2006|p=}}{{Equation box 1|indent=:|title=Incompressible Euler equations(convective or Lagrangian form)|equation=left{ begin{align} end{align}right.|cellpadding|border|border colour = #0073CF|background colour=#F5FFFA}}where the additional variables are:

• rho is the fluid mass density,
• p is the pressure, p = rho w.

The first equation, which is the new one, is the incompressible continuity equation. In fact the general continuity equation would be:{partial rho overpartial t} + mathbf u cdot nabla rho + rho nabla cdot mathbf u = 0but here the last term is identically zero for the incompressibility constraint.

Conservation form

{{See also|conservation equation|}}The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:mathbf y = begin{pmatrix}rho rho mathbf u end{pmatrix}; qquad {mathbf F} = begin{pmatrix}rho mathbf u rho mathbf u otimes mathbf u + p mathbf Imathbf uend{pmatrix}.Here mathbf y has length N+2 and mathbf F has size (N+2)N.{{efn|In 3D for example mathbf y has length 5, mathbf I has size 3×3 and mathbf F has size 5×3, so the explicit forms are:{mathbf y}=begin{pmatrix}rho rho u_1 rho u_2 rho u_3 end{pmatrix}; quad{mathbf 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 u_1 & u_2 & u_3 end{pmatrix}.}}In general (not only in the Froude limit) Euler equations are expressible as:frac {partial}{partial t}begin{pmatrix}rho rho mathbf u end{pmatrix}+ nabla cdot begin{pmatrix}rho mathbf urho mathbf u otimes mathbf u + p mathbf I mathbf uend{pmatrix} = begin{pmatrix}0 rho mathbf g 0 end{pmatrix}

Conservation variables

The variables for the equations in conservation form are not yet optimised. In fact we could define:{mathbf y}=begin{pmatrix}rho mathbf j end{pmatrix}; qquad {mathbf F}=begin{pmatrix} mathbf j frac {1} rho , mathbf j otimes mathbf j+ p mathbf I frac mathbf j rho end{pmatrix}.where mathbf j = rho mathbf u is the momentum density, a conservation variable.{{Equation box 1|indent=:|title=Incompressible Euler equation(s)(conservation or Eulerian form)|equation=frac {partial}{partial t}begin{pmatrix}rho mathbf j end{pmatrix}+ nabla cdot begin{pmatrix}mathbf j frac 1 rho , mathbf j otimes mathbf j + p mathbf I frac mathbf j rhoend{pmatrix} = begin{pmatrix}0 mathbf f 0 end{pmatrix}|cellpadding|border|border colour = #0073CF|background colour=#F5FFFA}}where mathbf f = rho mathbf g is the force density, a conservation variable.

Euler equations

In differential convective form, the compressible (and most general) Euler equations can be written shortly with the material derivative notation:{{Equation box 1|indent=:|title=Euler equations(convective form)|equation=left{begin{align} end{align}right.|cellpadding|border|border colour = #FF0000|background colour = #ECFCF4}}where the additional variables here is:

• e is the specific internal energy (internal energy per unit mass).

The equations above thus represent conservation of mass, momentum, and energy: the energy equation expressed in the variable internal energy allows to understand the link with the incompressible case, but it is not in the simplest form.Mass density, flow velocity and pressure are the so-called convective variables (or physical variables, or lagrangian variables), while mass density, momentum density and total energy density are the so-called conserved variables (also called eulerian, or mathematical variables).{{sfn|Toro|1999|p= 24}}If one expands the material derivative the equations above are:left{begin{align} end{align} right.

Incompressible constraint (revisited)

Coming back to the incompressible case, it now becomes apparent that the incompressible constraint typical of the former cases actually is a particular form valid for incompressible flows of the energy equation, and not of the mass equation. In particular, the incompressible constraint corresponds to the following very simple energy equation:frac{D e}{D t} = 0 Thus for an incompressible inviscid fluid the specific internal energy is constant along the flow lines, also in a time-dependent flow. The pressure in an incompressible flow acts like a Lagrange multiplier, being the multiplier of the incompressible constraint in the energy equation, and consequently in incompressible flows it has no thermodynamic meaning. In fact, thermodynamics is typical of compressible flows and degenerates in incompressible flows.{{sfn|Quartapelle|Auteri|2013|p=13|loc=Ch. 9}}Basing on the mass conservation equation, one can put this equation in the conservation form:{partial rho e over partial t} + nabla cdot (rho e mathbf u) = 0 meaning that for an incompressible inviscid nonconductive flow a continuity equation holds for the internal energy.

Enthalpy conservation

Since by definition the specific enthalpy is:h = e + frac p rho The material derivative of the specific internal energy can be expressed as:{D e over Dt} = {D h over Dt} - frac 1 rho left({D p over Dt} - frac p rho {D rho over Dt} right)Then by substituting the momentum equation in this expression, one obtains:{D e over Dt}= {D h over Dt} - frac 1 rho left(p nabla cdot mathbf u + {D p over Dt} right)And by substituting the latter in the energy equation, one obtains that the enthalpy expression for the Euler energy equation:{D h over Dt} = frac 1 rho {D p over Dt} In a reference frame moving with an inviscid and nonconductive flow, the variation of enthalpy directly corresponds to a variation of pressure.

Thermodynamics of ideal fluids

In thermodynamics the independent variables are the specific volume, and the specific entropy, while the specific energy is a function of state of these two variables.{{hidden|Deduction of the form valid for thermodynamic systems|Considering the first equation, variable must be changed from density to specific volume. By definition: Thus the following identities hold: Then by substituting these expressions in the mass conservation equation: And by multiplication: This equation is the only belonging to general continuum equations, so only this equation have the same form for example also in Navier-Stokes equations.On the other hand, the pressure in thermodynamics is the opposite of the partial derivative of the specific internal energy with respect to the specific volume:p(v, s) = - {partial e(v, s) over partial v}since the internal energy in thermodynamics is a function of the two variables aforementioned, the pressure gradient contained into the momentum equation should be explicited as:- nabla p (v,s) = - frac {partial p}{partial v} nabla v - frac {partial p}{partial s} nabla s = frac {partial^2 e}{partial v^2} nabla v + frac {partial^2 e}{partial v partial s}nabla s It is convenient for brevity to switch the notation for the second order derivatives: Finally, the energy equation:{D e over Dt} = - p v nabla cdot mathbf u can be further simplified in convective form by changing variable from specific energy to the specific entropy: in fact the first law of thermodynamics in local form can be written:{D e over Dt} = T {D s over Dt} - p {D v over Dt}by substituting the material derivative of the internal energy, the energy equation becomes:T {D s over Dt} + frac p {rho^2} left( {D rho over Dt} + rho nabla cdot mathbf u right) = 0now the term between parenthesis is identically zero according to the conservation of mass, then the Euler energy equation becomes simply:{D s over Dt} = 0|style = border: 1px solid lightgray; width: 90%;|headerstyle = text-align:left;}}For a thermodynamic fluid, the compressible Euler equations are consequently best written as:{{Equation box 1|indent=:|title=Euler equations(convective form, for a thermodynamic system)|equation=left{begin{align} end{align}right.|cellpadding|border|border colour = #FFFF00|background colour = #ECFCF4}}where:

• v is the specific volume
• mathbf u is the flow velocity vector
• s is the specific entropy

In the general case and not only in the incompressible case, the energy equation means that for an inviscid thermodynamic fluid the specific entropy is constant along the flow lines, also in a time-dependent flow. Basing on the mass conservation equation, one can put this equation in the conservation form:{{sfn|Landau|Lifshitz|2013|p=4|loc= Eqs 2.6 and 2.7}}{partial rho s over partial t} + nabla cdot (rho s mathbf u) = 0 meaning that for an inviscid nonconductive flow a continuity equation holds for the entropy.On the other hand, the two second-order partial derivatives of the specific internal energy in the momentum equation require the specification of the fundamental equation of state of the material considered, i.e. of the specific internal energy as function of the two variables specific volume and specific entropy:e = e(v, s)The fundamental equation of state contains all the thermodynamic information about the system (Callen, 1985),{{sfn|Henderson|2000|p=152|loc=2.6 Thermodynamic properties of materials}} exactly like the couple of a thermal equation of state together with a caloric equation of state.

Conservation form

{{See also|Conservation equation|}}The Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:mathbf y = begin{pmatrix} where:

• mathbf j = rho mathbf u is the momentum density, a conservation variable.
• E^t = rho e + frac{1}{2} rho u^2 is the total energy density (total energy per unit volume).

Here mathbf y has length N + 2 and mathbf F has size N(N + 2).{{efn|In 3D for example y has length 5, I has size 3×3 and F has size 3×5, so the explicit forms are: }} In general (not only in the Froude limit) Euler equations are expressible as:{hide}Equation box 1

{edih}where mathbf f = rho mathbf g is the force density, a conservation variable.We remark that also the Euler equation even when conservative (no external field, Froude limit) have no Riemann invariants in general.{{sfn|Chorin|Marsden|2013|p=118|loc=par. 3.2 Shocks}} Some further assumptions are requiredHowever, we already mentioned that for a thermodynamic fluid the equation for the total energy density is equivalent to the conservation equation:{partial over partial t} (rho s) + nabla cdot (rho s mathbf u) = 0 Then the conservation equations in the case of a thermodynamic fluid are more simply expressed as:{hide}Equation box 1|indent=:|title=Euler equation(s)(conservation form, for thermodynamic fluids)|equation= |cellpadding|border|border colour = #FFFF00|background colour = #ECFCF4}}where S = rho s is the entropy density, a thermodynamic conservation variable.Another possible form for the energy equation, being particularly useful for isobarics, is: where H^t = E^t + p = rho e + p + frac{1}{2} rho u^2 is the total enthalpy density.

Quasilinear form and characteristic equations

Expanding the fluxes can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to the Riemann problem. In regions where the state vector y varies smoothly, the equations in conservative form can be put in quasilinear form: where mathbf A_i are called the flux Jacobians defined as the matrices: Obviously this Jacobian does not exist in discontinuity regions (e.g. contact discontinuities, shock waves in inviscid nonconductive flows). If the flux Jacobians mathbf A_i are not functions of the state vector mathbf y, the equations reveals linear.

Characteristic equations

The compressible Euler equations can be decoupled into a set of N+2 wave equations that describes sound in Eulerian continuum if they are expressed in characteristic variables instead of conserved variables.In fact the tensor A is always diagonalizable. If the eigenvalues (the case of Euler equations) are all real the system is defined hyperbolic, and physically eigenvalues represent the speeds of propagation of information.{{sfn|Toro|1999|p= 44|loc=par 2.1 Quasi-linear Equations}} If they are all distinguished, the system is defined strictly hyperbolic (it will be proved to be the case of one-dimensional Euler equations). Furthermore, diagonalisation of compressible Euler equation is easier when the energy equation is expressed in the variable entropy (i.e. with equations for thermodynamic fluids) than in other energy variables. This will become clear by considering the 1D case.If mathbf p_i is the right eigenvector of the matrix mathbf A corresponding to the eigenvalue lambda_i, by building the projection matrix:mathbf{P} = left[mathbf{p}_1, mathbf{p}_2, ..., mathbf{p}_nright]One can finally find 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 the characteristic equations:{{sfn|Toro|1999|p= 52|loc= par 2.3 Linear Hyperbolic System}}frac{partial w_i}{partial t} + lambda_j frac{partial w_i}{partial r_j} = 0_iThe original equations have been decoupled into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds. The variables wi are called the characteristic variables and are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In one spatial dimension it is:w_i(x, t) = w_ileft(x - lambda_i t, 0right)Then the solution in terms of the original conservative variables is obtained by transforming back:mathbf{y} = mathbf{P} mathbf{w},this computation can be explicited as the linear combination of the eigenvectors:mathbf{y}(x, t) = sum_{i=1}^m w_ileft(x - lambda_i t, 0right) mathbf p_i,Now it becomes apparent that the characteristic variables act as weights in the linear combination of the jacobian eigenvectors. The solution can be seen as superposition of waves, each of which is advected independently without change in shape. Each i-th wave has shape w'i'p'i and speed of propagation λ'i. In the following we show a very simple example of this solution procedure.

Waves in 1D inviscid, nonconductive thermodynamic fluid

If one considers Euler equations for a thermodynamic fluid with the two further assumptions of one spatial dimension and free (no external field: g = 0):left{begin{align} end{align}right.If one defines the vector of variables:mathbf{y} = begin{pmatrix}v u send{pmatrix}recalling that v is the specific volume, u the flow speed, s the specific entropy, the corresponding jacobian matrix is:{mathbf A}=begin{pmatrix}u & -v & 0 - e_{vv} v & u & - e_{vs} v 0 & 0 & u end{pmatrix}.At first one must find the eigenvalues of this matrix by solving the characteristic equation:det(mathbf A(mathbf y) - lambda(mathbf y) mathbf I) = 0that is explicitly:detbegin{bmatrix}u-lambda & -v & 0 - e_{vv} v & u-lambda & - e_{vs} v 0 & 0 & u-lambda end{bmatrix}=0This determinant is very simple: the fastest computation starts on the last row, since it has the highest number of zero elements.(u-lambda) det begin{bmatrix}u-lambda & -v - e_{vv} v & u -lambda end{bmatrix}=0Now by computing the determinant 2×2:(u - lambda)left((u - lambda)^2 - e_{vv} v^2right) = 0by defining the parameter:a(v,s) equiv v sqrt {e_{vv}} or equivalently in mechanical variables, as:a(rho,p) equiv sqrt {partial p over partial rho} This parameter is always real according to the second law of thermodynamics. In fact the second law of thermodynamics can be expressed by several postulates. The most elementary of them in mathematical terms is the statement of convexity of the fundamental equation of state, i.e. the hessian matrix of the specific energy expressed as function of specific volume and specific entropy: is defined positive. This statement corresponds to the two conditions:left{begin{align} end{align}right.The first condition is the one ensuring the parameter a is defined real.The characteristic equation finally results:(u - lambda)left((u - lambda)^2 - a^2right) = 0That has three real solutions:lambda_1(v,u,s) = u-a(v,s) quad lambda_2(u)= u, quad lambda_3(v,u,s) = u+a(v,s)Then the matrix has three real eigenvalues all distinguished: the 1D Euler equations are a strictly hyperbolic system.At this point one should determine the three eigenvectors: each one is obtained by substituting one eigenvalue in the eigenvalue equation and then solving it. By substituting the first eigenvalue λ1 one obtains:begin{pmatrix}a & -v & 0 - e_{vv} v & a & - e_{vs} v 0 & 0 & a end{pmatrix} begin{pmatrix}v_1 u_1 s_1 end{pmatrix}=0Basing on the third equation that simply has solution s1=0, the system reduces to:begin{pmatrix}a & -v -a^2 /v& a end{pmatrix} begin{pmatrix}v_1 u_1 end{pmatrix}=0The two equations are redundant as usual, then the eigenvector is defined with a multiplying constant. We choose as right eigenvector: The other two eigenvectors can be found with analogous procedure as: Then the projection matrix can be built: Finally it becomes apparent that the real parameter a previously defined is the speed of propagation of the information characteristic of the hyperbolic system made of Euler equations, i.e. it is the wave speed. It remains to be shown that the sound speed corresponds to the particular case of an isentropic transformation:a_s equiv sqrt {left({partial p over partial rho} right)_s}

Compressibility and sound speed

Sound speed is defined as the wavespeed of an isentropic transformation:a_s(rho,p) equiv sqrt {left({partial p over partial rho} right)_s} by the definition of the isoentropic compressibility:K_s (rho,p) equiv rho left({partial p over partial rho} right)_s the soundspeed results always the square root of ratio between the isentropic compressibility and the density:a_s equiv sqrt {frac {K_s} rho}

Ideal gas

The sound speed in an ideal gas depends only on its temperature:a_s (T) = sqrt {gamma frac T m} {{hidden|Deduction of the form valid for ideal gases|In an ideal gas the isoentropic transformation is described by the Poisson's law:dleft(prho^{-gamma}right)_s = 0where γ is the heat capacity ratio, a constant for the material. By explicitating the differentials:rho^{-gamma} (d p)_s + gamma p rho^{-gamma-1} (d rho)_s =0and by dividing for ρ−γ dρ:left({partial p over partial rho}right)_s = gamma p rhoThen by substitution in the general definitions for an ideal gas the isentropic compressibility is simply proportional to the pressure:K_s (p) = gamma p and the sound speed results (Newton–Laplace law):a_s (rho,p) = sqrt {gamma frac p rho} Notably, for an ideal gas the ideal gas law holds, that in mathematical form is simply:p = n T where n is the number density, and T is the absolute temperature, provided it is measured in energetic units (i.e. in joules) through multiplication with the Boltzmann constant. Since the mass density is proportional to the number density through the average molecular mass m of the material: The ideal gas law can be recast into the formula: By substituting this ratio in the Newton–Laplace law, the expression of the sound speed into an ideal gas as function of temperature is finally achieved.|style = border: 1px solid lightgray; width: 90%;|headerstyle = text-align:left;}}Since the specific enthalpy in an ideal gas is proportional to its temperature:h = c_p T = frac {gamma}{gamma-1} frac T m the sound speed in an ideal gas can also be made dependent only on its specific enthalpy:a_s (h) = sqrt {(gamma -1) h}

Bernoulli's theorem for steady inviscid flow

Bernoulli's theorem is a direct consequence of the Euler equations.

Incompressible case and Lamb's form

The vector calculus identity of the cross product of a curl holds: where the Feynman subscript notation nabla_F is used, which means the subscripted gradient operates only on the factor mathbf F.Lamb in his famous classical book Hydrodynamics (1895), still in print, used this identity to change the convective term of the flow velocity in rotational form:{{sfn| Valorani| Nasuti|n.d.|pp= 11–12}}mathbf u cdot nabla mathbf u = frac{1}{2}nablaleft(u^2right) + (nabla times mathbf u) times mathbf uthe Euler momentum equation in Lamb's form becomes: {{See also|Cauchy momentum equation#Lamb form}}Now, basing on the other identity:nabla left( frac {p}{rho} right) = frac {nabla p}{rho} - frac{p}{rho^2} nabla rho the Euler momentum equation assumes a form that is optimal to demonstrate Bernoulli's theorem for steady flows:nabla left(frac{1}{2}u^2 + frac{p}{rho}right) - mathbf g = -frac{p}{rho^2} nabla rho + mathbf u times (nabla times mathbf u) - frac{partial mathbf u}{partial t} In fact, in case of an external conservative field, by defining its potential φ:nabla left( frac 1 2 u^2 + phi + frac p rho right) = -frac{p}{rho^2} nabla rho + mathbf u times (nabla times mathbf u) - frac{partial mathbf u}{partial t}In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:nabla left( frac 1 2 u^2 + phi + frac p rho right) = -frac{p}{rho^2} nabla rho + mathbf u times (nabla times mathbf u)And by projecting the momentum equation on the flow direction, i.e. along a streamline, the cross product disappears because its result is always perpendicular to the velocity:mathbf u cdot nabla left(frac{1}{2}u^2 + phi + frac{p}{rho}right) = -frac{p}{rho^2} mathbf u cdot nablarhoIn the steady incompressible case the mass equation is simply:mathbf u cdot nabla rho = 0,that is the mass conservation for a steady incompressible flow states that the density along a streamline is constant. Then the Euler momentum equation in the steady incompressible case becomes:mathbf u cdot nabla left( frac 1 2 u^2 + phi + frac p rho right) = 0The convenience of defining the total head for an inviscid liquid flow is now apparent: which may be simply written as:mathbf u cdot nabla b_l = 0That is, the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant.

Compressible case

In the most general steady (compressible) case the mass equation in conservation form is: mathbf{u} cdot nabla left({frac{1}{2}}u^2 + phi + frac{p}{rho}right) = frac{p}{rho}nabla cdot mathbf{u}The right-hand side appears on the energy equation in convective form, which on the steady state reads:mathbf u cdot nabla e = - frac{p}{rho} nabla cdot mathbf u The energy equation therefore becomes:mathbf u cdot nabla left( e + frac p rho + frac 1 2 u^2 + phi right) = 0, so that the internal specific energy now features in the head.Since the external field potential is usually small compared to the other terms, it is convenient to group the latter ones in the total enthalpy: and the Bernoulli invariant for an inviscid gas flow is: which can be written as:mathbf u cdot nabla b_g = 0That is, the energy balance for a steady inviscid flow in an external conservative field states that the sum of the total enthalpy and the external potential is constant along a streamline.In the usual case of small potential field, simply:mathbf u cdot nabla h^t sim 0

Friedmann form and Crocco form

{{See also|Crocco's theorem}}By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form:v nabla p = -T nabla s + nabla hin the convective form of Euler momentum equation, one arrives to:frac{Dmathbf u}{Dt}=T nabla,s-nabla ,hFriedmann deduced this equation for the particular case of a perfect gas and published it in 1922.{{sfn|Friedmann|1934|p=198|loc=Eq 91}} However, this equation is general for an inviscid nonconductive fluid and no equation of state is implicit in it.On the other hand, by substituting the enthalpy form of the first law of thermodynamics in the rotational form of Euler momentum equation, one obtains:frac{partialmathbf{u}}{partial t} + frac{1}{2} nablaleft(u^2right) + (nabla times mathbf{u}) times mathbf{u} + frac{nabla p}{rho} = mathbf{g}and by defining the specific total enthalpy:h^t = h + frac{1}{2}u^2one arrives to the Crocco–Vazsonyi form{{sfn|Henderson|2000|p=177|loc=par. 2.12 Crocco's theorem}} (Crocco, 1937) of the Euler momentum equation:frac{partial mathbf{ u}}{partial t} + (nabla times mathbf u) times mathbf u - T nabla s + nabla h^t = mathbf{g}In the steady case the two variables entropy and total enthalpy are particularly useful since Euler equations can be recast into the Crocco's form:left{begin{align} end{align}right.Finally if the flow is also isothermal:T nabla s = nabla (T s)by defining the specific total Gibbs free energy: the Crocco's form can be reduced to:left{begin{align} end{align}right.From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow.

Discontinuities

{{See also|Shock waves|Burgers equation}}The Euler equations are quasilinear hyperbolic equations and their general solutions are waves. Under certain assumptions they can be simplified leading to Burgers equation. 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 equations. 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.To properly compute the continuum quantities in discontinuous zones (for example shock waves or boundary layers) from the local forms{{efn|Sometimes the local and the global forms are also called respectively differential and non-differential, but this is not appropriate in all cases. For example, this is appropriate for Euler equations, while it is not for Navier-Stokes equations since in their global form there are some residual spatial first-order derivative operators in all the caractheristic transport terms that in the local form contains second-order spatial derivatives.}} (all the above forms are local forms, since the variables being described are typical of one point in the space considered, i.e. they are local variables) of Euler equations through finite difference methods generally too many space points and time steps would be necessary for the memory of computers now and in the near future. In these cases it is mandatory to avoid the local forms of the conservation equations, passing some weak forms, like the finite volume one.

Rankine–Hugoniot equations

Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain:nabla cdot mathbf F = mathbf 0 where in general F is the flux matrix. By integrating this local equation over a fixed volume Vm, it becomes: Then, basing on the divergence theorem, we can transform this integral in a boundary integral of the flux: This global form simply states that there is no net flux of a conserved quantity passing through a region in the case steady and without source. In 1D the volume reduces to an interval, its boundary being its extrema, then the divergence theorem reduces to the fundamental theorem of calculus: that is the simple finite difference equation, known as the jump relation: That can be made explicit as: where the notation employed is: Or, if one performs an indefinite integral: On the other hand, a transient conservation equation:{partial y over partial t} + nabla cdot mathbf F = mathbf 0 brings to a jump relation: For one-dimensional Euler equations the conservation variables and the flux are the vectors:mathbf y = begin{pmatrix} frac{1}{v} j E^t end{pmatrix}, mathbf F = begin{pmatrix} j v j^2 + p v j left(E^t + pright) end{pmatrix}, where:

• v is the specific volume,
• j is the mass flux.

In the one dimensional case the correspondent jump relations, called the Rankine–Hugoniot equations, are: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.{{sfn|Babinsky|2003}} Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem".{{sfn|Imai|1973|p=}}This "theorem" explains clearly why there are such low pressures in the centre of vortices,{{sfn|Babinsky|2003}} which consist of concentric circles of streamlines.This also is a way to intuitively explain why airfoils generate lift forces.{{sfn|Babinsky|2003}}

Exact solutions

All potential flow solutions are also solutions of the Euler equations, and in particular the incompressible Euler equations when the potential is harmonic.{{sfn|Marchioro|Pulvirenti |1994|p=33}}(File:OS schematic.svg|thumb|right|300px|A two-dimensional parallel shear-flow.)Solutions to the Euler equations with vorticity are:

• parallel shear flows – where the flow is unidirectional, and the flow velocity only varies in the cross-flow directions, e.g. in a Cartesian coordinate system (x,y,z) the flow is for instance in the x-direction – with the only non-zero velocity component being u_x(y,z) only dependent on y and z and not on x.{{sfn|Friedlander |Serre|2003|p=298}}
• Arnold–Beltrami–Childress flow – an exact solution of the incompressible Euler equations.
• Two solutions of the three-dimensional Euler equations with cylindrical symmetry have been presented by Gibbon, Moore and Stuart in 2003.{{sfn|Gibbon |Moore |Stuart|2003|p=}} These two solutions have infinite energy; they blow up everywhere in space in finite time.

See also

• Bernoulli's theorem
• Kelvin's circulation theorem
• Cauchy equations
• Froude number
• Madelung equations
• Navier–Stokes equations
• Burgers equation
• Jeans equations
• Perfect fluid

References

Notes

{{Notelist}}

Citations

{{Reflist}}

Sources

• BOOK, Anderson, John, Computational Fluid Dynamics,weblink 1995, McGraw-Hill Education, 978-0-07-001685-9,
• {{citation | journal=Physics Education | first=Holger | last=Babinsky | date=November 2003 | url=http://www.iop.org/EJ/article/0031-9120/38/6/001/pe3_6_001.pdf | title=How do wings work? | volume=38 | issue=6 | pages=497–503 | doi=10.1088/0031-9120/38/6/001 | bibcode=2003PhyEd..38..497B | s2cid=1657792 }}
• BOOK, Chorin, Alexandre J., Marsden, Jerrold E., A Mathematical Introduction to Fluid Mechanics,weblink 2013, Springer, 978-1-4612-0883-9,
• JOURNAL, 10.1090/S0273-0979-07-01181-0, Christodoulou, Demetrios, The Euler Equations of Compressible Fluid Flow, Bulletin of the American Mathematical Society, 44, 4, October 2007,weblink 581–602, free,
• JOURNAL, Mémoires de l'académie des sciences de Berlin, 11, 1757, 274–315, Principes généraux du mouvement des fluides, Leonhard, Euler,weblink fr, The General Principles of the Movement of Fluids,
• BOOK, Fay, James A., Introduction to Fluid Mechanics,weblink 1994, MIT Press, 978-0-262-06165-0,
• BOOK, Handbook of Mathematical Fluid Dynamics – Volume 2, Friedlander, S., Serre, D., 978-0-444-51287-1, 2003, Elsevier,
• BOOK, Friedmann, A., Alexander Friedmann, An essay on hydrodynamics of compressible fluid, Опыт гидромеханики сжимаемой жидкости, ru, Saint Petersburg, Petrograd, 1922, 1934,weblink Nikolai Kochin, Nikolai, Kochin,
• JOURNAL, J.D., Gibbon, D.R., Moore, J.T., Stuart, J. T. Stuart, Exact, infinite energy, blow-up solutions of the three-dimensional Euler equations, Nonlinearity, 16, 5, 1823–1831, 2003, 10.1088/0951-7715/16/5/315, 2003Nonli..16.1823G, 250797052,
• BOOK, Ben-Dor, Gabi, Igra, Ozer, Elperin, Tov, Handbook of Shock Waves, Three Volume Set,weblink 2000, Elsevier, 978-0-08-053372-8, L.F., Henderson, General Laws for the Propagation of Shock-waves through Matter,
• {{citation |title=An Introduction to the Incompressible Euler Equations |first=John K. |last=Hunter |date=25 September 2006 |url=https://www.math.ucdavis.edu/~hunter/notes/euler.pdf |access-date=2019-05-31 }}
• BOOK, 今井 功 (IMAI, Isao), 『流体力学(前編)』, Fluid Dynamics 1, 裳華房 (Shoukabou), November 1973

  language=Japanese

  Imai, 1973, }} BOOK, Landau, L D, Lifshitz, E. M., Fluid Mechanics,weblink 2013, Elsevier, 978-1-4831-4050-6, BOOK, Mathematical Theory of Incompressible Nonviscous Fluids, Marchioro, C., Pulvirenti, M., 0-387-94044-8, Applied Mathematical Sciences, 96, 1994, Springer, New York, BOOK, Quartapelle, Luigi, Auteri, Franco, Fluidodinamica comprimibile,weblink 2013, CEA, it, 978-88-08-18558-7, Compressible Fluid Dynamics, BOOK, Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction,weblink 1999, Springer, 978-3-540-65966-2, {{citation|publisher=Sapienza - Universit`a di Roma|title=Metodi di analisi delle turbomacchine|first1=Mauro|last1=Valorani|first2=Francesco|last2=Nasuti|url=http://web2srv.ing.uniroma1.it/~m_valorani/GasTurbines_LM_files/DispenseTurboMacchine.pdf|access-date=2019-05-31|date=n.d.|archive-date=2022-05-16|archive-url=https://web.archive.org/web/20220516185800weblink|url-status=dead}} {{citation|first=M.|last=Zingale|title=Notes on the Euler equations|date=16 April 2013|url=http://bender.astro.sunysb.edu/hydro_by_example/compressible/Euler.pdf|access-date=2019-05-31|archive-date=2015-06-19|archive-url=https://web.archive.org/web/20150619141500weblink|url-status=dead}} Further reading BOOK, 978-3-319-59694-5, Badin, G., Crisciani, F., Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws -, Springer, 2018, 218, 10.1007/978-3-319-59695-2, 2018vffg.book.....B, 125902566, BOOK, G. K., Batchelor, An Introduction to Fluid Dynamics, 1967, Cambridge University Press, 0-521-66396-2, BOOK, Philip A., Thompson, 1972, Compressible Fluid Flow, McGraw-Hill, New York, 0-07-064405-5, {{Leonhard Euler}}{{Authority control}}

  
  
  
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