Theoretical and Numerical Issues of Incompressible Fluid .... Frey, chap3.pdf · The mathematical analysis of incompressible Navier-Stokes equations for viscous ﬂows has been a

Theoretical and Numerical Issues of Incompressible Fluid Flows

Chapter 3: The Navier-Stokes Model

Instructor: Pascal Frey

Sorbonne Université, CNRS4, place Jussieu, Paris

University of Tehran, 2018

A classical example

• Von Karman vortex street: flow around a cylinder

Figure 1: Von Karman vortex street at various Reynolds numbers.

Theoretical and numerical issues for Incompressible Fluid Flows University of Tehran, April-May, 2018 2/ 60

A classical example


Figure 2: Re

A classical example


Figure 3: Re ∼ 10 steady laminar flow (streamlines of velocity).


A classical example


Figure 4: Re ∼ 100 Karman vortex street (streamlines of velocity).


A classical example


Figure 5: Re ∼ 1e5 laminar flow with largely unsteady and turbulent wake (Streamlines of turbulent kineticenergy).


A classical example


Figure 6: Re ∼ 1e6 steady fully turbulent flow field (streamlines of velocity).


Section 3.1Mathematical and numerical analysis

The Navier-Stokes model

• Suppose the fluid is confined in a domain Ωt ⊂ Rd (i.e. an open and connected regionin two or three dimensions of space), for t ∈ [0, T ].

• We recall that the isothermal flow of Newtonian viscous fluids of constant density ismodeled by the following Navier-Stokes equations:

Given f, find u such that

ρ

(∂u

∂t+ (u · ∇)u

)− µ∆u+∇p = ρf , in Ωt × (0, T ) ,

div u = 0 , in Ωt × (0, T ) ,

(1)

where u represents the velocity [m/s], p the pressure [Pa = n/m2], µ is the dynamicviscosity [kg/(ms)] ρ is the density [kg/m3] and f denotes a density of volume forces permass unit [N/m3].

• The non-linear term (u · ∇)u corresponds to the convective acceleration, while thediffusion term ∆u models the viscous effects.



• By dividing both sides of the first equation by the constant density ρ, we obtain thealternate formulation

Given f, find u such that∂u

∂t+ (u · ∇)u− ν∆u+

1

ρ∇p = f , in Ωt × (0, T ) ,

div u = 0 , in Ωt × (0, T ) ,

(2)

where ν = µ/ρ is the kinematic viscosity coefficient [m2/s].

• Introducing the Reynolds number Re = UL/ν = ρUL/µ, where L and U are char-acteristic of the length and velocity scales of the flow, and dividing both sides of (2) byU2/L, allows us to rewrite the equations in a dimensionless form:

∂u

∂t+ (u · ∇)u−

1

Re∆u+∇p = f , in Ωt × (0, T ) ,

div u = 0 , in Ωt × (0, T ) .(3)

where the introduced the change of variables

x = x/L , t = t(U/L) , u = u/U , p = p/ρU2 , f = f(L/U2) .



• solving the Navier-Stokes equations is a nontrivial task for the following reasons:◦ the momentum equation is nonlinear◦ the incompressibility condition div u = 0◦ solving the Navier-Stokes equation amounts to solve a system of (d+ 1 if Ω ⊂ Rd)PDEs coupled through the nonlinear term (u · ∇)u, the incompressibility conditiondiv u = 0 and sometimes through the viscous term and the boundary condition;

• a time discretization by operator splitting will partly overcome these difficulties, in par-ticular to decouple the difficulties associated to the nonlinearity whith those associatedto the incompressibility condition.

• the numerical solution of time dependent PDEs by operator-splitting methods has beenlargely studied by researchers, but the specialists of ODEs have so far shown little inter-est in splitting methods.


Initial and boundary conditions


• As such, problems (1) or (3) are not well-posed since these equationsmay have an infinitenumber of solutions.

• It is mandatory to introduce further conditions, namely(a) an initial condition (i.e. at time t = 0):

u(x,0) = u0(x) , in Ω (4)

where u0 is a given divergence-free vector field and(b) a Dirichlet boundary condition:

u(x, t) = g(x, t) , on ∂Ω× (0, T ) , (5)

where g(x, t) is a given function defined over ∂Ω× (0, T ).If Ω is bounded, then we have, thanks to the divergence theorem and the continuitycondition, the compatibility condition:∫

∂Ωg(x, t) · nds =

∫Ω

div u(x, t) dx = 0 , (6)

where n is the outward unit normal to the domain boundary ∂Ω.


Initial and boundary conditions

• Other boundary conditions, for example(c) mixed boundary conditions:

u = g0 , on ΓD × (0, T ) σ n = g1 , on ΓN × (0, T ) (7)

ΓD and ΓN are such that Γ̊D ∩ Γ̊N = ∅, ΓD ∪ΓN = ∂Ω, g0 and g1 are two givenfunctions of (x, t).σ is the stress tensor: σ = 2µD(u)− p Idand D(u) is the deformation rate tensor: 2D(u) = ∇u+ (∇u)t.

(d) In applications, other mixed conditions are often preferred:

u = g0 , on ΓD × (0, T ) µ∂u

∂n− p n = g1 , on ΓN × (0, T ) (8)

◦ Notice that in most applications, the function g1 is set to zero in the Neumannboundary condition.


Variational formulation

• We return to the Navier-Stokes equations in primitive variables:


∂t+ (u · ∇)u− ν∆u+

1

ρ∇p = f , in Ωt × (0, T ) ,

div u = 0 , in Ωt × (0, T ) ,u = u0 , with div u0 = 0 ,

(9)

endowed with the following mixed boundary conditions:

u = g0 , on ΓD × (0, T ) σ n = g1 , on ΓN × (0, T )

where we considered σ = 2νD(u)−p

ρId . (10)

• The mathematical theory for the analysis of the Navier-Stokes model and the estimationof the approximation error in finite element method rely on functional analysis andSobolev spaces.

• In particular, the finite element approximation requires to recast the Navier-Stokes equa-tions into a weak form.


Weak form of the problem

• We introduce the Hilbert functional space

V = {v ∈ H10(Ωt) ; v = 0 on ΓD} ,

which is a Hilbert space for the scalar product and the norm

(u, v)H1(Ω) =∫

Ωu(x)v(x) dx+

∫Ω∇u(x) : ∇v(x) dx

‖u‖H1(Ω) =∫

Ω|u(x)|2 dx+

∫Ω|∇u|2 dx

• Space V will coincide with H10(Ωt) if the Dirichlet boundary ΓD coincide exactly with∂Ωt. In this case, the prescribed Dirichlet function g0 must be compatible with thedivergence-free constraint, i.e.∫

∂Ωtg0 · nds =

∫Ωt

div udx = 0 .



• By multiplying the first equation by a test function v ∈ V and integrating over Ωt, weobtain for almost every t ∈ (0, T ):∫

Ωt

∂u

∂t·v dx+

∫Ωt

((u·∇)u)·v dx−∫

Ωtν∆u·v dx+

1

ρ

∫Ωt∇p·v dx =

∫Ωtf ·v dx .

• Green’s formula allows to rewrite the partial terms as follows

−∫

Ωtν∆u · v dx =

∫Ωtν∇u : ∇v dx−

∫∂Ωt

ν∂u

∂nds∫

Ωt∇p · v dx =−

∫Ωtpdiv v dx+

∫∂Ωt

p v · nds

• Then, taking into account the mixed boundary conditions (10), we have for almost everytime t in (0, T )∫

Ωt

∂u

∂t· v dx+

∫Ωt

((u · ∇)u) · v dx+∫

Ωtν∇u : ∇v dx−

1

ρ

∫Ωtpdiv v dx

=∫

Ωtf · v dx+

∫ΓN

g1 · v ds , ∀ v ∈ V (11)

completed with the Dirichlet boundary condition (10) on ΓD × (0, T ).Theoretical and numerical issues for Incompressible Fluid Flows University of Tehran, April-May, 2018 16/ 60


• As expected, the Neumann boundary condition σ n = g1 on ΓN × (0, T ) is naturallysatisfied by the formulation, which is often called the variational formulation of themomentum equation.

• Introducing the deformation rate tensor D(u) yields the following formulation, for al-most any t ∈ (0, T ):∫

Ωt

∂u

∂t· v dx+

∫Ωt

((u · ∇)u) · v dx+ 2ν∫

ΩtD(u) : D(v) dx−

1

ρ

∫Ωtpdiv v dx

=∫

Ωtf · v dx+

∫ΓN

g1 · v ds , ∀ v ∈ V (12)

completed with the Dirichlet boundary condition (10) on ΓD × (0, T ).Likewise, the Neumann boundary condition is satisfied automatically.

• Similarly for the continuity equation, we first multiply the equation by a test function qin a space Q and then we integrate it over Ωt, to obtain the equation:∫

Ωtq div udx = 0 , ∀ q ∈ Q . (13)


Existence of solution to the problem

Remark 1. (i) The main difference between this weak formulation and the Stokes weak for-mulation is related to the convective term. The latter is identified with a trilinear formc : H1(Ωt)3 → R that will be defined later.

(ii) All integrals involved in the bilinear and the trilinear forms are finite.

• The mathematical analysis of incompressible Navier-Stokes equations for viscous flowshas been a topic of interest for almost a century since they were introduced by Navierin 1822 and Stokes in 1845.

• The seminal work of Leray proved the existence of solutions when the flow region is thefull space Rn.

• An early worthwhile result is due to J.L. Lions established that if the flow domain is two-dimensional then the unsteady Navier-Stokes equations lead a unique solution.

• In three dimensions of space, there is no theoretical result about the uniqueness (andregularity) of a solution to the time- dependent Navier-Stokes equations modelling theflow of incompressible viscous fluids.


The steady Navier-Stokes problem

• We first consider the steady Navier-Stokes equations that describe the motion of anhomogeneous incompressible Newtonian fluid.

• The motion is here independent of time.

• The problem is posed in a bounded domainΩ ⊂ Rd with a Lipschitz continuous bound-ary ∂Ω and reads as follows:

Given f, find u such that

− ν∆u+ (u · ∇)u+1

ρ∆p = f , in Ω

div u = 0 , in Ω

(14)

In addition, we consider homogeneous Dirichlet boundary conditions,

u = 0 , on ∂Ω , (15)

corresponding to the case where the fluid is confined in the domain with fixed boundary.

• This problem close to the Stokes problem, except for the presence of the non-linearconvective term (u · ∇)u) that makes it much more difficult to solve.


Mathematical results

• Let consider the functional spaces V = H10(Ω) and Q = L20(Ω).

• The variational formulation is almost identical to the formulation (11), except for theintegral of the time derivative:Given f ∈ L2(Ω), find (u, p) such that for all v ∈ V∫

Ω((u ·∇)u) ·v dx+

∫Ων∇u : ∇v dx−

1

ρ

∫Ωpdiv v dx =

∫Ωf ·v dx+

∫ΓN

g1 ·v ds

(16)

• The weak formulation of the continuity equation is strictly identical to equation (13):∫Ωq div udx = 0 , ∀ q ∈ Q . (17)

• The formulation (16) can be conveniently recast using bilinear and trilinear forms.



• We introduce the bilinear forms a(·, ·) : (u, v) ∈ V × V → a(u, v) ∈ R and b(·, ·) :(u, q) ∈ V ×Q→ b(u, q) ∈ R, (same as the Stokes problem):

a(u, v) =∫

Ων∇u : ∇v dx =

∫Ων

d∑i=1

∇ui : ∇vi dx , (18)

b(u, q) = −∫

Ωdiv u q dx . (19)

and the trilinear form c(·; ·, ·) : (u,w, v) ∈ V × V × V → c(w;u, v) ∈ R associatedwith the non-linear term and defined as:

c(w;u, v) =∫

Ω((w · ∇)u) · v dx =

d∑i,j=1

∫Ωwj∂ui∂xj

vi dx . (20)

• We have already stated that the bilinear forms a(·, ·) and b(·, ·) are continuous.

• The next two propositions give useful properties of the trilinear form c(·; ·, ·).



• We have the following results:Proposition 1. The trilinear form c(·; ·, ·) is continuous on (H1(Ω))d.

• For any given u ∈ V , the map v 7→ c(u, u, v) is linear and continuous on V .•

Proposition 2. Let u, v, w in H1(Ω) and let assume divw = 0 and w · n|∂Ω = 0.Then, the trilinear form c : (H1(Ω))3 → R satisfies the following properties, which areequivalent:

c(w; v, v) = 0 , (21)c(w;u, v) + c(w; v, u) = 0 . (22)

• Assume the constant density ρ = 1. We can recast the problem (16):given f ∈ L2(Ω), find u ∈ V , p ∈ Q such that{

a(u, v) + c(u;u, v) + b(v, p) = (f, v) , for all v ∈ Vb(u, q) = 0 , for all q ∈ Q

. (23)

• A pair of function (u ∈ V, p ∈ Q) satisfying these equations, given f ∈ L2(Ω), is aweak solution of the stationary Navier-Stokes problem.


The nonlinear term

different forms of the nonlinear term

• (u · ∇)u : convective form

• div(uut) : divergence form

• div(uut) + 12∇(utu) : divergence form with modified pressure

• (∇× u)× u : rotational form (with modified pressure)

the equivalence of these forms can be established.

• from (u · ∇)u one can obtain:

(u · u)u+∇p = div(uut) +1

2∇(utu)−

1

2∇(utu) +∇p

= div(uut) +1

2(utu) +∇pmod with pmod = p−

1

2(utu) ;

• the convective form and the divergence form are trivially equivalent since one gets:

div(uut) = (u · ∇u) + (div u)u = (u · ∇)u .


Properties of the convective term

Lemma 1 (basic properties). • let v be weakly differentiable, then it is

(u · ∇)v = (∇v)u

• let u be weakly differentiable, then it holds

(u · ∇)u = D(u)u+1

2(div u)× u

• the variational form of the convective term is trilinear, i.e., it is linear in each argument;

• let u, v, w ∈ H1(Ω), then

((u · ∇)v, w) = (div(vut), w)− ((div u)v, w) ,((u · ∇)v, w) = (u,∇(v · w))− ((u · ∇)w, v) ,((u · ∇)v, w) = ((∇v)tw, u)

((u · ∇)v, w) =∫∂Ω

(u · n)(v · w) ds− (div u, v · w)− ((u · ∇)w, v) ,

where n is the outward unit normal to ∂Ω.

• let u, v ∈ H1(Ω), w ∈ Hdiv then

((∇× u)× v, w) = ((v · ∇)u,w)− ((w · ∇)u, v) .


Convective term

Lemma 2 (estimates of the convective term). Let u, v, w ∈ H1(Ω), where Ω ⊂ Rd is abounded domain with Lipschitz boundary, then there is a C ∈ R such that

((u · ∇)v, w) ≤ C‖u‖H1(Ω)‖∇v‖L2(Ω)‖w‖H1(Ω) ,

and from the divergence form of the convective term

((u · ∇)v, w) +1

2((div u)v, w) ≤ C‖u‖H1(Ω)‖∇v‖H1(Ω)‖w‖H1(Ω) ,

If w ∈ Vdiv then it is for the rotational form

((∇× u)× v, w) ≤ C‖u‖L2(Ω)‖∇v‖H1(Ω)‖w‖H1(Ω) .

Lemma 3 (estimate of the convective term). letu ∈ L2(Ω), v ∈W1,∞(Ω) andw ∈ H1(Ω)then

((u · ∇)v, w) ≤ ‖u‖L2(Ω)‖∇v‖L∞(Ω)‖w‖L2(Ω) .



• Let consider the following divergence-free functional subspaces:

Vdiv = {v ∈ H1(Ω) ; div v = 0} , and V 0div = {v ∈ Vdiv ; v = 0 on ΓD} ,

• Problem (23) has an equivalent form: find (u, p) ∈ V0

div × L20(Ω) such that

a(u, v) + c(u;u, v) + b(v, p) = (f, v) , for all v ∈ H10(Ω) .(24)

If (u, p) is a solution to problem (23), then u is a solution to problem (24). The converseis also true as stated next.

•

Lemma 4. Let Ω be a bounded domain of Rd with a Lipschitz-continuous boundary ∂Ω.Given f ∈ L2(Ω), there exists at least one pair (u, p) ∈ Vdiv×L20(Ω) which satisfiesproblem (24) or equivalently problem (23).



• Let consider the norm of the trilinear form (c·; ·, ·) of V 3 and recall the norm of V ′:

supu,v,w∈V

c(w;u, v)

|u|H1(Ω)|v|H1(Ω)|w|H1(Ω)= C , ‖f‖V ′ = sup

v∈V

(f, v)

|v|H1(Ω).

• The next result gives the uniqueness of a weak solution (u, p) to the steady problem(24).

•

Theorem 1. Under the hypothesis of the previous lemma, and assuming that

C

v2‖f‖V ′ ≤ 1 ,

then, problem (24) has a unique solution (u, p) in V × L20(Ω).


Boundary conditions

• From the numerical point of view, we have to consider two types of boundary conditions:(i) artificial conditions: a virtual boundary is considered, e.g. by introducing a wall to

account for the symmetry of the flow and reduce the size of half;(ii) physical conditions: a boundary exists in the domain and must be treated accord-

ingly, e.g. no-slip condition at wall.

• The homogeneous Dirichlet boundary condition u = 0, (no-slip), is the correct andnatural condition for a viscous fluid contained in a rigid domain. Indeed, the viscouseffect constraints the fluid particles to adhere to the wall.

• When Dirichlet conditions u = g are specified on the whole boundary ∂Ω, the pressuresolution is known up to a hydrostatic constant only and then the function g has tosatisfies a comptability condition: ∫

∂Ωg · nds = 0 .


Boundary conditions

• For specific applications or in different situations, more appropriate boundary condi-tions need to be worked out.

• The problem of well-posed boundary conditions is thus a key issue in many fields.◦ Slip boundary condition: the influence of the wall rugosity on the slip behavior of thefluid is still a matter of debate. In case of impermeable boundary, there is a classicalcondition for the normal velocity component:

u · n = 0 , on ∂Ω

◦ For the tangential behavior however, the situation is less clear. A complete slipboundary condition has been proposed:

u · n = [D(u) · n]tan = 0 , on ∂Ω ,

which means that the fluid is confined in the domain but the fluid particle can slipalong the boundary.


Boundary conditions

• Additional boundary conditions have been proposed:◦ Neumann non-friction boundary conditions: used to prescribe a force per unit areathat involves the normal component of the stress tensor σ:

σn = (2µD(u)− p)n = g1 , on ΓN ,

where ΓN is a subset of ∂Ω.When g1 = 0, this condition is usually used as an free outflow condition on afictitious boundary, e.g. symmetry wall, or at the interfaces, e.g. free surfaces orbetween two fluids. In this case, the part of the boundary ΓN is called a free outflow.In other situations, g1 = −pen, where pe denotes the external pressure.

◦ alternate boundary conditions and formulations for the viscous term −ν∆u: whenthe coefficient ν is constant, the viscous term can be rewritten in different equivalentforms:

ν∆u = div(ν(∇u+ (∇u)t)) (25)= −ν curl(curlu) = ν(∇(div u)− curl(curlu)) . (26)


Boundary conditions

• Suppose Γn and Γτ are two subsets of ∂Ω, such that Γ̊n ∩ Γ̊τ = ∅, Γn ∪ Γτ = ∂Ω.

• We want to prescribe the normal and the tangent velocities on Γn and Γτ , respectively:

u · n = gn , on Γn and n× (u× n) = gτ , on Γτ , (27)

where n denotes the outward normal vector to the domain boundary, u · n is the com-ponent of u normal to the boundary and n × u × n = u − (u · n)n represents theprojection of u onto the tangent (plane) to the boundary.

• We introduce the functional spaces:

Vg = {v ∈ H1(Ω); v · n = gn on Γn , n× v × n = gτ on Γτ}V0 = {v ∈ H1(Ω) ; v · n = 0 on Γn , v × n = 0 on Γτ

and we consider here Q = L20(Ω) if Γn = ∂Ω) or Q = L2(Ω) otherwise.


Boundary conditions

• The weak formulation of problem (14) reads:find (u, p) ∈ Vg ×Q such thata(u, v) + c(u;u, v) + b(v, p) = (f, v) + d(v) , for all v ∈ V0b(u, q) = 0 , for all q ∈ Q

. (28)

Here, we have introduced in the right-hand side a linear functional d(·) defined as:

d(v) =∫∂Ω\Γn

r v · nds+∫∂Ω\Γτ

s · v × nds ,

where the functions r and s have a physical meaning.

• Taking for a(·, ·) the form (18), the natural boundary conditions become

−p+ νn · ∇u · n = r , on ∂Ω\Γn , (29)νn · ∇u× n = s , on ∂Ω\Γτ (30)


Boundary conditions

• As such, these conditions have no physical meaning, and the boundary conditions (27)must be imposed on all boundaries.

• For more pratical conditions, e.g. for free-surface problems or artifical outflow bound-aries, one may prefer dealing with the formulation (25) for the viscous term associatedwith the bilinear form:

a(u, v) =1

2

∫ΩνD(u) : D(v) dx , where 2D(u) = ∇u+ (∇u)t .

And the natural boundary conditions write:

−p+ 2νn ·D(u) · n = r , on ∂Ω\Γn , (31)2νn ·D(u)× n = s , on ∂Ω\Γτ (32)

where r and s are then the normal and tangent stresses, respectively.


Non-linear iterative procedures

model: the main difference with Stokes equations is the presence of nonlinear terms in the(stationary) Navier-Stokes equations.

• we need iterativemethods to solve incompressible flows; the literature contains a varietyof such methods which are all based on a linearization of the equations.

• given an initial guess, u0 ∈ H1(Ω), a sequence of iterates {uk}k=1,...,n ∈ H1(Ω) iscomputed which is expected to converge towards the solution of the weak formulation.

• the fixed-point (Picard) linearization of the stationary Navier-Stokes equations (14) leadsto a sequence of so-called Oseen problems, i.e., linear problems (endowed with bound-ary conditions (14)) of the form

− ν∆u+ (w · ∇)u+cu+∇p = f , in Ω ,div u = 0 , in Ωu = 0 , on ∂Ω

(33)

where w ∈ V is the approximation of the solution from the previous Picard iteration.


Weak solution of Oseen problem

• the weak form of Oseen problem reads as follows:given w ∈ V , find (u, p) ∈ V ×Q such that

ν(∇u,∇v) + ((w · ∇)u+cu, v)− (div v, p) = (f, v) ∀v ∈ V−(div u, q) = 0 ∀q ∈ Q

(34)

Theorem 2 (Existence and Uniqueness of a Weak Solution of the Oseen Equations).LetΩ be a bounded domain in Rd, d = 2,3, with a Lipschitz continuous boundary ∂Ω;and let the conditions on the data of the Oseen problem be fulfilled (w in Ld+1(Ω)),u, v ∈ Lq(Ω). Then, there exists a unique solution u, p ∈ H10(Ω)× L

20(Ω) of (34).

Lemma 5 (stability of the solution). under the conditions of the theorem, the solutionof the Oseen problem (34) depends continuously on the data of the problem:

ν

2‖∇u‖2

L2(Ω)+‖c1/2u‖2

L2(Ω) ≤1

2ν‖f‖2

H−1(Ω)‖∇u‖

if f ∈ L2(Ω) and c(x) ≥ c0 > 0, then we have also the stability bound

ν‖∇u‖2L2(Ω)+

1

2‖c1/2u‖2

L2(Ω) ≤1

2c0‖f‖2

L2(Ω)‖∇u‖


Oseen iteration

• the weak formulation of the Oseen problem is:given w ∈ V , find (u, p) ∈ V ×Q such that{

a(u, v) + c(w, u, v) + b(v, p) = (f, v) , for all v ∈ Vb(u, q) = 0 for all q ∈ Q .

• the Oseen-iteration is a secant modulus method, where the equations are linearized byfreezing the nonlinearity: the first term in c(·, u, v) is evaluated at the last iterate;

Algorithm 1 (Oseen iteration). :for k = 0,1,2, . . .

given uk ∈ V, pk ∈ Q, solve (uk+1, pk+1)a(uk+1, v) + c(uk, uk+1, v) + b(v, pk+1) = (f, v) , for all v ∈ Vb(uk+1, q) = 0 for all q ∈ Q .

(35)

• At each iteration step k, one linear system (35) has now to be solved.


Oseen iteration

• note that the previous result does not depend on the size of the data;

• however, if the data are larger the numerical solution of the Oseen equation is diffi-cult (similar to the Navier–Stokes equations with large Reynolds number) and it is notguaranteed that the nonlinear iteration converges.

• the convergence of the Oseen–Iteration can be shown for small data using the Banachfixed point theorem:

Theorem 3. if ν2 ≥ C‖f‖L2(Ω), the iterate uk obtained from the Oseen method con-

verge to the unique solution (u, ) ∈ V ×Q of the Navier-Stokes problem.

• in general, the rate of convergence of Oseen iteration is linear.


Newton method

• the Newton method seems appealing because of its theoretical potential to achievequadratic convergence rate.

• consider the abstract problem of finding the root x̃ of a general nonlinear function f ,f(x) = 0.

• suppose xk is an approximate solution of the root x̃.

• then, by a Taylor expansion around an initial guess xk we write

0 = f(x̃) = f(xk) + f′(xk)(x̃− xk) + o(|x̃− xk|)

≈ f(xk) + f ′(xk)(x̃− xk) ,(36)

and thus

x̃ ≈ xk − [f ′(xk)]−1 f(xk) .

• Newton’s method for solving f(x) = 0 writes:

xk+1 = xk − [f ′(xk)]−1 f(xk) , k = 0,1,2, . . .


Newton’s method

• likewise, let define the function for the Navier-Stokes model:

f(u,∇u) = (u · ∇)u .

• in this case, a Taylor expansion of f(·,∇·) around uk,∇uk reads

fk+1(u,∇u) ≈ fk(uk,∇uk) + (uk+1 − uk) ·∂fk∂u

+∇(uk+1 − uk) ·∂fk∂∇u

.

• neglecting the quadratic terms and substituting the definition of f(·,∇·) yields

uk+1 · ∇uk+1 ≈ uk · ∇uk + δk · ∇uk +∇δk · uk= uk+1 · ∇uk + uk · ∇uk+1 − uk · ∇uk ,

(37)

which forms the kernel of the classical Newton linearization of the nonlinear term.

• The convergence rate is quadratic; if ‖u− u0‖ ≤ δ

‖uk+1 − u‖ ≤ M‖uk − u‖2 , and ‖uk − u‖ ≤(Mδ)2k

M,


Newton’s method

• Newton’s method applied to the weak formulation of the stationary Navier-Stokes prob-lem and using identity (37) takes the following form:

Algorithm 2 (Newton-method).

for k = 0,1,2, . . .1. given uk ∈ V, solve δuk ∈ V

a(δuk, v) + c(uk, δuk, v) + c(δuk, uk, v) =

−(a(uk, v) + c(uk, uk, v)− (fk, v)

)2. update uk+1 = uk + δuk ;

(38)

• an important aspect for Newton-type methods is the starting value, since one only ob-tains local convergence in general.

• one possibility is to use some steps of an Oseen-iteration first, and to use the result asa starting value for a Newton method.


Section 3.2Discretization procedures

Unsteady Navier-Stokes equations

• We return to the unsteady Navier-Stokes equations:


∂t+ (u · ∇)u− ν∆u+

1

ρ∇p = f , in Ωt × (0, T ) ,

div u = 0 , in Ωt × (0, T ) ,u = u0 , with div u0 = 0 ,

(39)

endowed with the conditions (to be well-posed):

u = g , on ∂Ω× (0, T ) with∫∂Ω

g(t) · nds = 0 , on (0, T ) (40)

• Then, we have for almost every time t in (0, T )∫Ωt

∂u

∂t· v dx+

∫Ωt

((u · ∇)u) · v dx+∫

Ωtν∇u : ∇v dx−

1

ρ

∫Ωtpdiv v dx

=∫

Ωtf · v dx , ∀ v ∈ V (41)

completed with the Dirichlet boundary condition on ∂Ω× (0, T ).


Temporal discretizations

• discretization in time:θ-schemes use the following strategy for the full discretization and linearization:1. semi-discretization of (41) in time. The semi-discretization in time leads in eachdiscrete time to a nonlinear system of equations of saddle point type.

2. variational formulation and linearization. The nonlinear system of equations is refor-mulated as variational problem and the nonlinear variational problem is linearized.

3. discretization of the linear systems in space. The linear system of equations arising ineach step of the iteration for solving the nonlinear problem is discretized by a finiteelement discretization using, e.g., an inf-sup stable pair of finite element spaces.

• this approach, which applies first the discretization in time and then in space, is alsocalled method of Rothe or horizontal method of lines. The other way, discretizing first inspace to get an ordinary differential equation and then in time, is called vertical) methodof lines.



one step θ-schemes for the Navier-Stokes equations

• let θ ∈ [0,1] and consider the time step from tn to tn+1.

• one step θ-schemes for the Navier-Stokes equations are of the form:

un+1

∆tn+1+ θ

(−ν∆un+1 + (un+1 · ∇)un+1

)+∇pn+1 =

un

∆tn+1− (1− θ) (−ν∆un + (un · ∇)un) + (1− θ)fn + θfn+1

and div un+1 = 0 (42)

• explicit and implicit Euler scheme, Crank–Nicolson scheme:three well known one-step θ-schemes are obtained by an appropriate choice of:◦ θ = 0: forward or explicit Euler scheme,◦ θ = 1/2: Crank–Nicolson scheme,◦ θ = 1: backward or implicit Euler scheme.



one step θ-schemes for the Navier-Stokes equations

• forward or explicit Euler scheme: the appearance of the viscous term leads to a stiffODE with respect to time. From the numerical analysis of ODEs, it is known that explicitschemes have to be used with very small time steps for stiff problems to obtain stablesimulations.For the Navier–Stokes equations, the time step has to be usually so small that the sim-ulations become very inefficient, hence not recommended for the discretization.

• backward or implicit Euler scheme: this first order scheme is quite popular. However,the use of the backward Euler scheme in combination with higher order discretizationsin space might lead to rather inaccurate results, compared with the results computedwith higher order temporal discretizations.

• Crank–Nicolson scheme: is a second order scheme whose use is very popular.



concerning the incompressibility constraint

• the temporal derivative is approximated in (42) at tn + θtn+1.Thus, the velocity is approximated at this time and the pressure acts as a Lagrangianmultiplier for this velocity.

• for instance, in the case of the Crank-Nicolson scheme, the computed solution approxi-mates (un+1/2, pn+1/2).

• if one is interested in a good approximation of the pressure at time tn+1, one has tomodify the pressure term in the Crank–Nicolson scheme to

1

2(pn+1 + pn) .

• however, this scheme requires in the first step the pressure at the initial time.


Discretization in space

• We perform the space discretization of the NS system as previously, to find (uh, ph) inVh ×Qh for all t ∈ (0, T ) such that:

∫Ωt

∂uh∂t· vh dx+

∫Ωt

((uh · ∇)uh) · vh dx+∫

Ωtν∇uh : ∇vh dx

−1

ρ

∫Ωtph div vh dx =

∫Ωtfh · vh dx , ∀vh ∈ V0h∫

Ωtdiv uhqh dx = 0 ∀qh ∈ Qh

uh = gh(t) on ∂Ωtuh(0) = u0h

• The finite element spaces Vh and Qh are classically

V0h = Vh ∩(H10(Ωt)

)2= {vh ∈ Vh ; vh = 0 on ∂Ωt} ,

• The function fh, u0h and gh are convenient approximations of f, u0 and g and gh mustverify ∫

∂Ωtgh(t) · nds = 0 on (0, T ) .


Operator splitting

We introduce a θ-scheme (Glowinski, 1985) to discretize an initial value problem:

∂u

∂t+A(u, t) = 0 , with u(0) = u0 .

• Let θ be a number in the open interval (0,1/2),

• the scheme applied to the solution of this problem, when A = A1 +A2, solves un+θ,un+1−θ and un+1

• it writes:

u0 = u0un+θ − un

θδt+A1(u

n+θ, (n+ θ)δt) +A2(un, nδt) = 0

un+1−θ − un+θ

(1− 2θ)δt+A1(u

n+θ, (n+ θ)δt) +A2(un+1−θ, (n+ 1− θ)δt) = 0

un+1 − un+1−θ

δt+A1(u

n+1, (n+ 1)δt) +A2(un+1−θ, (n+ 1− θ)δt) = 0


Time discretization procedures

• The full discretization of unsteady Navier-Stokes equations by a θ-scheme leads to thefollowing systems:◦ initializations and parameter settings:u

0h = u0hθ = 1− 1/

√2 , α = (1− 2θ)/(1− θ) , β = θ/(1− θ) ,

◦ and then, unh being know,

◦ compute first (un+θh , pn+θh ) ∈ Vh ×Qh by solving the elliptic systems, for n ≥ 0:

∀vh ∈ V0h,

∫Ωt

un+θh − unh

θδt· vh dx+ αν

∫Ωt∇un+θh : ∇vh dx−

∫Ωtpn+θh div vh dx

=∫

Ωtfn+θh · vh dx− βν

∫Ωt∇unh : ∇vh dx−

∫Ωt

(unh · ∇)unh · vh dx∫

Ωtdiv un+θh qh = 0 , ∀qh ∈ Qh

un+θh = gn+θh on ∂Ωt



◦ compute un+1−θh ∈ Vh, for all vh ∈ V0h

∫Ωt

un+1−θh − un+θh

(1− 2θ)δt· vh dx+ βν

∫Ωt∇un+1−θh : ∇vh dx

+∫

Ωt(un+1−θh · ∇)u

n+1−θh · vh dx

=∫

Ωtfn+θh · vh dx− αν

∫Ωt∇un+θh : ∇vh dx−

∫Ωtpn+θh div vh dx

un+1−θh = gn+1−θh on ∂Ωt ,



◦ and finally compute (un+1h , pn+1h ) ∈ Vh ×Qh

∫Ωt

un+1h − un+1−θh

(θ)δt· vh dx+ αν

∫Ωt∇un+1h : ∇vh dx−

∫Ωtpn+1h div vh dx

=∫

Ωtfn+1h · vh dx− βν

∫Ωt∇un+1−θh : ∇vh dx

−∫

Ωt(un+1−θh · ∇)u

n+1−θh · vh dx∫

Ωtdiv un+1h qh dx = 0 ∀qh ∈ Qh

un+1h = gn+1h on ∂Ωt ,


Discretization of the total derivative

• Until now, the term (u · ∇)u was treated as a general nonlinearity;

• However, in Navier-Stokes equations, the term

ρ

(∂u

∂t+ (u · ∇)u

)models the transport of the momentum ρu by the velocity field u.

• The idea of combining FE approximations and methods of characteristics for numericalsimulations of incompressible viscous fluid flows is quite old (1970’s): J.P. Benque, O.Pironneau, Gresho, . . . but still considered a bit "exotic".

• Despite the simplicity of the principle that underlies themethods of characteristics, theirpractical implemntation is delicate and not fully understood;

• "Any method for approximating hyperbolic equations sacrifices a good deal if it takes noaccount of the method of characteristics" (Morton, 1992),"Numerical schemes that follow characteristics backward in time and then interpolateat their feet have a history stretching back in the very early days of computational fluiddynamics" (Courant et al., 1952).


Solution of transport problems

• We introduce the method of characteristics to solve a transport problem:∂ϕ

∂t+ u · ∇ϕ = 0 in Ω× (0, T )

ϕ(0) = ϕ0

ϕ = g on Γ− × (0, T )

(43)

with∂u

∂t= 0 div u = 0

∂g

∂t= 0

and

Γ− = {x ∈ ∂Ω, u(x) · n(x) < 0} .

• Let consider (x∗, t∗) ∈ Ω× (0, T ); we associate the solution of the systemdX

dt= u(x)

X(t∗) = x∗

the solution of this system is X(·;x∗, t∗)


Solution of transport problems

• The curve Cx∗,t∗ described by the points {X(t;x∗, t∗), t} as t varies is called the char-acteristic curve associated to the transport equation (43) and to {x∗, t∗}.

• Suppose ϕ is solution of (43) and restrict it to the curve Cx∗,t∗, we have then

d

dtϕ(X(t;x∗, t∗), t) =

(∂ϕ

∂t+ u · ∇ϕ

)(X(t;x∗, t∗), t) = 0 .

• this shows that ϕ is constant along the curve Cx∗,t∗.

• it follows that if ϕ solve (43), we have, for τ > 0 sufficiently small, the relation

ϕ(x, t) = ϕ(X(t− τ ;x, t), t− τ) .



• The backward method of characteristics can be used to solve Navier-Stokes problem:◦ initialization

u0 = u0

◦ and for n ≥ 0:un+1 − un∗

δt− ν∆un+1 +∇pn+1 = fn+1 in Ωt

div un+1 = 0 , in Ωtun+1 = gn+1 on ∂Ωt ;

(44)

◦ where un∗ is obtained by solving the system:dX

dt= un(X) on (0, δt)

X(δt) = x(45)

we denote by Xn(·, x) the solution of the problem (45).◦ We take un∗(x) = un(Xn(0, x))


Section 3.3Numerical examples

2d lid-driven cavity

• classical test case: simulations for Reynolds number ranging from 400 up to 10000.◦ problem involves a primary vortex at the cavity center and vortices in the corners,◦ number of vortices increases with Re and position of the center of primary vortexmoves toward the center,

◦ 4 meshes have been used: a regular triangulation (2,461 nodes, 5,000 elements);a uniform triangulation (2,143 nodes, 4,136 elements); a refined uniform trian-gulation (8,421 nodes, 16,544 elements); and a regular triangulation (10,201nodes, 20,000 elements);

◦ results in good accordance with experimental results (Ghia et al.).

Re = 400 Re = 1,000 Re = 5,000 Re = 10,000


Backfacing step

• we consider a two-dimensional channel flow over a downstream-facing step.

• it consists in an upstream channel of height h followed by a suddenly enlarged channelof height H = 2h.

• the channel lengths of the portions situated upstream and downstream of the backstepare l0H and l1H , respectively.

• at the upstream end of the channel there is a fully developed laminar flow defined bythe steady axial velocity

u(y) = 24u0y(0.5− y) , for x = −l0 and y ∈ [0,0.5]

where u0 represents the mean axial flow velocity.


Backfacing step

• in the unsteady flow case, a portion of the lower wall of lengthHl, situated just behindthe downstream-facing step, is assumed to execute transverse oscillations defined bythe following lower wall equation:

Hy =

{Hg(x, t)− hs for x ∈ [0, l]− hs for x > l

where g(x, t) is a sinusoidal-shape mode defined as

g(x, t) = e(t) sin(πx/l) where e(t) = A cos(ωt)


Backfacing step

• in the unsteady flow case, a portion of the lower wall of lengthHl, situated just behindthe downstream-facing step, is assumed to execute transverse oscillations.

Backward facing step: Backward facing step

reprinted from [Mateescu D., J. Fluids and Structures, 2001]


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