5

The Incompressible Euler and Navier–Stokes
Equations

Various liquids carried by virtue of their own weight from various points to
form a pool of various liquids at a point of accumulation. (Lawrence Weiner,
conceptual artist describing in words a piece of art instead of realizing it
physically, 1978)

In a reasonable theory there are no dimensionless numbers whose values are
only empirically determinable. (Einstein)

5.1 The Incompressible Euler Equations

We now specialize the Euler equations (4.1) for ˆ

u = (ρ, u, e) to incompressible

flow assuming constant density ρ = 1:

∇ · u = 0

in Q,

˙u + (u

· ∇)u + ∇p = f

in Q,

˙e +

∇ · (eu + pu) = 0

in Q,

u

· n = 0

on Γ

× I,

u(

·, 0) = u

0

in Ω,

e(

·, 0) = e

0

in Ω,

(5.1)

where we write the momentum equation in vector form noting that by incom-
pressibility and constant unit density

∇·(m

i

u) = u

·∇u

i

+ u

i

∇·u = (u·∇)u

i

.

The equation of mass conservation and the momentum equations give four
equations for the four unknowns (u, p). Formally we can thus solve for (u, p)
in the first two equations and then separately solve for the total energy e with
(u, p) given.

We thus start out considering the reduced system in (u, p) which we will

also refer to as the Euler equations for incompressible flow:

˙u + (u

· ∇)u + ∇p = f

in Q,

∇ · u = 0

in Q,

u

· n = 0

on Γ

× I,

u(

·, 0) = u

0

in Ω,

(5.2)

44

5 The Incompressible Euler and Navier–Stokes Equations

where to conform to common notational practice we changed the order of the
equations for mass conservation and momentum.

5.2 The Incompressible Navier–Stokes Equations

We now generalize to the Navier–Stokes equations including viscous forces
modeled in the form of a Laplacian term

−ν∆u in the momentum equation,

where ν > 0 is a constant coefficient of viscosity. We thus assume the fluid to
be Newtonian with viscous stresses depending linearly on velocity gradients.
We shall below scrutinize the general concept of viscosity and the particu-
lar assumption of a Newtonian fluid. For now we assume that the viscous
force is modeled by the term

−ν∆u in the momentum equation, with further

justification in Section 28.3.

The Navier–Stokes equations (NS) for a Newtonian fluid enclosed in a

fixed volume Ω in

R

3

read:

˙u + (u

· ∇)u − ν∆u + ∇p = f,

in Q,

∇ · u = 0,

in Q,

u = 0,

on Γ

× I,

u(

·, 0) = u

0

,

in Ω,

(5.3)

where now all components of the velocity u are prescribed to be zero on the
boundary Γ , not only the normal velocity u

· n as in the Euler equations. The

boundary condition u = 0 is referred to as a no slip boundary condition or ho-
mogeneous Dirichlet boundary condition for the velocity u. We consider other
boundary conditions below including inflow and outflow boundary conditions
and friction boundary conditions.

5.3 What is Viscosity?

In fact, viscosity closely couples to turbulence and thus the viscosity in NS
would rather represent a turbulent viscosity ν = ν(u) with a possibly very
complex dependence on the velocity u; with a non-constant ν the viscous
term would rather take the form

−∇ · (ν∇u). But if indeed ν depends on

u and we cannot determine how, then the NS equations would seem to be
useless for predictions, even using computational methods.

So how are we going to handle this problem? Well, first we recall that we

focus on the case of small viscosity. The first idea that comes up is then of
course to assume that the viscosity is so small that we can put it equal to
zero, which gives us the Euler equations. This is the classical approach of Euler
which however led to d’Alembert’s Mystery, and thus had to be abandoned
as if the great scientist Euler was incorrect. We will show that Euler was not
stupid at all; in fact Euler’s solution to the tricky problem of finding a proper

5.3 What is Viscosity?

45

(small) viscosity by simply setting the viscosity to zero, is very elegant and
far-reaching. But to come to this satisfactory conclusion we have to solve the
Euler equations computationally, and not try analytical methods.

Next we assume that ν is non-vanishing, but still small, so that we are

facing NS equations. We shall below see that solving NS equations computa-
tionally we have to use a mesh with a certain mesh size h, and we shall see
that the computational method itself introduces an effect which can be viewed
as a mesh dependent artificial viscosity ν

h

. Now, if ν

h

> ν, then the true vis-

cosity ν will be overshadowed by the artificial viscosity ν

h

, which means that

the precise value of ν becomes irrelevant. And the smaller ν is, the bigger is
the chance that it will be overshadowed by the artificial viscosity ν

h

, so that

we do not have to determine ν accurately; it would be sufficient just to know
that ν is (sufficiently) small.

Of course, we expect the solution (u, p) = (u

ν

, p

ν

) to depend on ν, and

in order for a computation with an artificial viscosity ν

h

> ν to have some

predictive value, it is necessary that the output or quantity of interest from
the solution (u

ν

, p

ν

) is not critically depending on the precise value of ν. We

shall below see that this may indeed be true in many cases if the output is
a mean value in space-time such as a drag coefficient c

D

(ν) measuring the

total force of a body moving through a fluid with viscosity ν. Thus, we shall
see that c

D

(ν) varies quite slowly with ν, which means that we do not have

to know ν very precisely (which certainly helps when determining ν), or that
we effectively can compute with an artificial viscosity ν

h

and only need to

know that ν < ν

h

. Thus, the good news is that for turbulent flow with small

viscosity, in many cases we do not need to specify the viscosity very precisely,
which would be very difficult.

We shall also study cases with a critical dependence on (small) viscosity,

including the so-called drag crisis reflecting that the resistance of e.g. a cylin-
der of unit diameter and speed moving through a fluid, quite suddenly drops
by more than 50% as ν decreases to about 10

−6

to raise again for smaller ν.

As indicated, we shall see that simply assuming ν = 0, in a case where we

know that the viscosity is small but not exactly how small, will take us quite
far. This follows the initial ingenious idea of Euler of studying ideal fluids
with zero viscosity, but we shall see that to arrive at this peak, we will have
to pass through the deep valley of d’Alembert’s Mystery.

To sum up: Turbulence occurs in fluids with small viscosity ν and typi-

cal outputs may have a weak dependence on ν. This means that we do not
need precise information on ν; in many cases we may effectively set ν = 0
following Euler, or knowing just one binary digit of ν may be enough. Thus
turbulent flow is difficult because of its complexity, but may be easy because
precise information on the viscosity is not needed. This is favorable for com-
putation, because complexity is handled by brute computational power, while
the viscosity advantage remains.

46

5 The Incompressible Euler and Navier–Stokes Equations

5.4 What is Heat Conductivity?

We have seen that in the Euler equations we assume no flow of heat by con-
duction, which we may express as setting the coefficient of heat conductivity
to zero. We will see that nevertheless heat will flow in turbulent fluid flow, not
by conduction but by convection with the fluid. In turbulent flow heat energy
thus will have a tendency to spread out (by turbulent convection), and if we
don’t see the turbulent motion of the fluid we may attribute the flow of heat
to some form of conduction with a certain coefficient of heat conductivity. We
understand that in turbulent flow the effective coefficient of heat conduction
may be a very complex function of the flow, and thus impossible to determine.

We will see in G2 simulations of the Euler equations that we can directly

compute the heat flow (by turbulent convection). This way we are thus re-
lieved from the difficult task of finding an effective heat conductivity, since
the objective of determining such a coefficient is to compute the heat flow.

To sum up, we are led to the conclusion that the classical need of determin-

ing coefficients of viscosity and heat conductivity in order to compute viscous
stresses and heat flow, disappears in turbulent flow with small viscosity and
heat conductivity, since the fluid motion and heat flow is directly computable
from the Euler equations. We believe this is an elegant mathematical solution
of a very difficult if not impossible practical problem.

5.5 Friction Boundary Conditions

We start our studies in Chapter 9 considering the Euler equations with slip
boundary conditions, following the historical development of fluid dynamics,
to discover several surprising facts. Later we shall also discuss friction bound-
ary conditions for the Euler equations with the tangential stress coupled to the
tangential velocity with a friction parameter, with slip corresponding to zero
friction. We will view the friction boundary conditions as a simple so-called
wall model for the flow in a turbulent boundary layer close to the boundary,
with the friction parameter depending on the Reynolds number.

We will find that we may simulate flows with very large Reynolds numbers

using the Euler equations with proper friction boundary conditions, without
computationally (fully) resolving the boundary layer. Computational solution
of the Euler equations thus will become extremely useful all along Euler’s
original plans, which resurrects Euler’s model after a long dark age of discredit
caused by d’Alembert’s Mystery.

5.6 Einstein’s Ideal

We note that the given data for NS equations is represented by (Ω, I, f, u

0

)

together with the viscosity ν, which we refer to as a parameter, and ˆ

u = (u, p)

5.7 Euler and NS as Dynamical Systems

47

is the corresponding solution which we seek. The specification of the data
(Ω, I, f, u

0

) is usually clear, while the determination of the viscosity ν (or

Reynolds number) is much less clear, as we have seen.

In the Euler equations we set the viscosity ν to zero (and heat conductivity

as well) with the intention to model fluid flow with very small viscosity. The
Euler equations represent the ideal of Einstein, namely a model without any
(dimensionless) parameter, such as the Reynolds number. We shall see below
that we may predict quantitative properties of fluid from the Euler equations
only supplying the data (Ω, I, f, u

0

) but not any viscosity. We may thus pre-

dict the drag of body in a flow with small viscosity by only supplying the
shape of the body as data! This is like predicting the circumference (= 2πr)
of a circle supplying only the radius r of the circle as data.

Einstein dreamed about forming his equations of general relativity without

any parameter, but was forced to put in a cosmological constant (to model a
static universe), which he declared was his “biggest blunder”. But the Euler
equations for incompressible flow is a true example of Einstein’s ideal model.

But is it really possible to make predictions about the flow of a fluid

without knowing the viscosity of the fluid, only knowing that the viscosity is
small? In the book we shall show that this is possible, but not in the classical
way that led to catastrophe assuming that the effect of viscosity was zero, but
in a new non-trivial way building on the observation that outputs of turbulent
flows may have a weak dependence on viscosity!

5.7 Euler and NS as Dynamical Systems

The Euler and NS equations are examples of a dynamical system of the general
form of an initial value problem ˙u(t) = g(u(t)) for t

∈ (0, ˆt ], u(0) = u

0

, where

g(v) is a given function of v, u

0

is a given initial value, and t

→ u(t) is the

solution defined on [0, ˆ

t ]. We say that the function v

→ g(v) expresses the law

of the dynamical system. We also refer to a solution t

→ u(t) as a trajectory

of the dynamical system. In a dynamical system of this form time changes
continuously over an interval of time from an initial time 0 to a final time ˆ

t.

If g is bounded, then a solution u(t) is continuous in time.