4

The Euler Equations

However sublime are the researches on fluids which we owe to Messrs
Bernoulli, Clairaut and d’Alembert, they flow so naturally from my two
general formulae that one cannot sufficiently admire this accord of their
profound meditations with the simplicity of the principles from which I
have drawn my equations ...(Euler 1752)

4.1 Foundation of Fluid Dynamics

Fluid dynamics as a scientific discipline starts with the Euler equations ex-
pressing conservation of mass, momentum and total energy as a system of
partial differential equations. We now formulate these equations for an invis-
cid fluid enclosed in a fixed (open) domain Ω in three-dimensional space

R

3

with boundary Γ over a time interval [0, ˆ

t ] with initial time zero and final

time ˆ

t. An inviscid fluid (viscosity ν = 0) is also said to be an ideal fluid.

We thus assume that there are no viscous forces (inviscid flow) and we also
assume that there is no heat flow from conduction (zero heat conductivity).

We seek the density ρ, momentum m = ρu, with u = (u

1

, u

2

, u

3

) the

velocity, and the total energy e as functions of (x, t)

∈ Ω ∪ Γ × [0, ˆt], where

x = (x

1

, x

2

, x

3

) denotes the coordinates in

R

3

and u

i

is the velocity in the

x

i

-direction. The Euler equations for ˆ

u

≡ (ρ, m, e) read with Q = Ω × I and

I = (0, ˆ

t ]:

˙

ρ +

∇ · (ρu) = 0

in Q,

˙

m

i

+

∇ · (m

i

u) + p

,i

= f

i

in Q,

i = 1, 2, 3,

˙e +

∇ · (eu + pu) = 0

in Q,

u

· n = 0

on Γ

× I,

ˆ

u(

·, 0) = ˆu

0

in Ω,

(4.1)

where p = p(x, t) is the pressure of the fluid, p

,i

= ∂p/∂x

i

is the partial

derivative with respect to x

i

, the dot indicates differentiation with respect

to time, n denotes the outward unit normal to Γ and f = (f

1

, f

2

, f

3

) is a

40

4 The Euler Equations

given volume force (like gravity) acting on the fluid, and ˆ

u

0

= ˆ

u

0

(x) represent

initial conditions. Further, the total energy e = k + θ, where k = ρ

|u|

2

/2 is

the kinetic energy, with

|u|

2

≡ u

2

1

+ u

2

2

+ u

2

3

, and θ = ρT is the internal energy

with T the temperature scaled so that c

v

= 1, where c

v

is the heat capacity

under constant volume.

The boundary condition u

· n = 0 is a slip boundary condition requiring

the normal velocity u

·n to vanish corresponding to an impenetrable boundary

with zero friction. Below we will consider other boundary conditions, including
inflow and outflow boundary conditions and non-zero friction.

Further,

∇ · v =

3

i=1

v

i,i

denotes the divergence of v = (v

1

, v

2

, v

3

), while the gradient of a scalar func-

tion w will be denoted by

∇w = (w

,1

, w

,2

, w

,3

).

There are five equations in the Euler system (4.1), while the number of

unknowns including the pressure is six, and so we need one more equation.
This equation may be a state equation for a compressible gas expressing the
pressure p as a function of density ρ and temperature T , e.g. the state equation
p = (γ

− 1)ρT of a perfect gas, where γ = c

p

is the adiabatic index with c

p

the

heat capacity under constant pressure, and (γ

− 1) is the gas constant. The

additional equation may alternatively express that the fluid is incompressible
in the form

∇ · u = 0 in Q.

For a perfect gas, the speed of sound c is given by c

2

= γ(γ

− 1)T , and

the Mach number is defined as M =

|u|/c, with u the velocity of the gas. We

assume in this book that the fluid is incompressible with constant density.
This is an approximation up to a variation in density of size M

2

, so that for

M < 0.3 the variation would be less than say 10 %. Traveling through air thus
corresponds to velocities less than 300 km/h (200 mph), which is applicable
to most cars, small airplanes, or even jumbo-jets at start and landing. Water
can be viewed to be incompressible in most applications.

The extension of the Euler equations to include viscous forces and heat

flow by conduction are referred to as the Navier–Stokes equations. The Navier–
Stokes and Euler equations describe a very rich complex world of fluid dy-
namics.

In this book we focus mainly on incompressible inviscid or viscous flow,

and open to compressible flow in the last chapter, which we continue in
Body & Soul Vol 5 on Computational Thermodynamics.

4.2 Derivation of the Euler Equations

We now show that the Euler equations (4.1) express conservation of mass,

momentum and total energy in the conservation variables (ρ, m, e). To this

4.3 The Euler Equations as a Continuum Model

41

end consider a fixed small volume V in Ω with boundary S. Mass conservation
implies that



V

˙

ρ dx =

∂

∂t



V

ρ dx =

−



S

(ρu)

· n ds,

expressing that the rate of increase of total mass in the fixed volume V is
equal to the rate of inflow through the boundary S. The Divergence Theorem
(see e.g. B&S Vol 3 [36]) states that



V

∇ · (ρu) dx =



S

(ρu)

· n ds,

and we thus conclude that



V

˙

ρ dx +



V

∇ · (ρu) dx = 0

for all volumes V . Assuming that the integrands are continuous, we thus
obtain the equation for mass conservation ˙

ρ +

∇ · (ρu) = 0.

We obtain the differential equation expressing conservation of each compo-

nent of the momentum m

i

similarly, noting that by Newtons 2nd law the rate

of change of momentum is given by the corresponding component

−p

,i

of the

pressure gradient

∇p with increasing pressure retarding the flow, combined

with a volume force f

i

. Finally, the equation expressing conservation of total

energy e is obtained noting that the rate of change of the total energy is given
by the work pu.

4.3 The Euler Equations as a Continuum Model

The Euler equations represent a continuum model with formally no smallest
scale, since there is no smallest scale of the set of real numbers

R. In reality we

solve the Euler equations in finite precision computation using the G2 finite
element method with finite mesh size h, which may vary in space and time. We
may think of the finite precision as effectively computing with a fixed number
of digits (e.g. double precision with about 16 digits) instead of computing
with real numbers with infinitely many digits with infinite precision (which
is impossible). Typical meshes have a mesh size of 10

−2

on the unit cube

with 10

6

mesh points. A gas has about 10

24

molecules per mole, and thus the

values of density, momentum and energy at each mesh point represent mean
values of about 10

18

molecules, thus mean values over incredibly many “fluid

particles”.

A computational particle model of a gas accounting for the position and

velocity of each of the 10

24

particles in each mole, is inconceivable on any

kind of thinkable computer. Thus, only continuum models can be used for
macroscopic phenomena of fluid flow.

42

4 The Euler Equations

4.4 Incompressible Flow

In an incompressible fluid the density ρ does not change if we follow the motion
of the fluid particles of the flow. We can express this fact in the differential
equation form

D

u

ρ

≡ ˙ρ + u · ∇ρ = 0,

where D

u

ρ is the convective derivative of ρ with respect to the velocity u. We

obtain the convective derivative by computing the change in time following
the trajectory x(t) of a fluid particle satisfying the differential equation ˙

x(t) =

u(x, t). Differentiating ρ(x(t), t) with respect to time, we obtain by the chain
rule:

d

dt

ρ(x(t), t) = ( ˙

ρ + ˙x

· ∇ρ)(x(t), t) = D

u

ρ(x(t), t).

Since mass conservation reads D

u

ρ + ρ

∇ · u = 0, we conclude that the

velocity u in incompressible flow is characterized by the equation

∇ · u = 0, in Q.

(4.2)