International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 2, Jun 2013, 15-22 © TJPRC Pvt. Ltd.
SEMI-IMPLICIT BACKWARD EULER SCHEME FOR THE NAVIER-STOKES 3D PROBLEMS: A REVIEW ONANAYE A. S.1 & ODEKUNLE M. R.2 1 2
Mathematical Sciences Department, Redeemer’s University, Redemption City, Mowe, Ogun State, Nigeria
Department of Mathematics, Federal University of Technology, (Now Called Modibbo Adama University of Technology), Yola, Nigeria
ABSTRACT In this paper, a review of Navier-Stokes (NS) problems was carried out particularly on Semi-implicit Backward Euler Scheme. Other areas of applications of NS were looked into and some of their methods of solutions as well. 2010 Mathematics Subject Classification: 35A01,35B35, 35Q86.
KEYWORDS: Navier-Stokes, Backward Euler, Semi-Implicit, Scheme, Discrete, Global Error and Symplectic Integrator
INTRODUCTION In Physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton’s second law to fluid motion , together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term. Landau and Lifshitz (1987). The equations are useful because they describe the physics of many things of academic and economic interest. They are used to model the weather ,ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell’s equations they can be used to model and study magneto-hydrodynamics. Acheson, (1990) and Batchelor,(1967). Emanuel (2001), the Navier–Stokes equations are of great interest in a purely mathematical sense. Surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularity (smoothness). These are called the Navier-Stokes existence and smoothness problems. The Navier–Stokes equations dictate not position but rather velocity. A solution of the Navier–Stokes equations is called a velocity field or flow field, which is a description of the velocity of the fluid at a given point in space and time. Once the velocity field is solved for, other quantities of interest (such as flow rate or drag force) may be found. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid; however for visualization purposes one can compute various trajectories.
PRELIMINARY: THE NATURE OF NAVIER–STOKES EQUATIONS The Navier–Stokes equations are nonlinear partial differential equations in almost every real situation. In some cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear
16
Onanaye A. S. & Odekunle M. R.
equations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model. Polyanin, Kutepov, Vyazmin, and Kazenin (2002) and Currie (1974). Though the flow may be steady (time independent), the fluid decelerates as it moves down the diverging duct (assuming incompressible flow), hence there is an acceleration happening over position. Convective acceleration is represented by the nonlinear quantity:
v.∇v which may be interpreted either as vector
v.∇ or as v.(∇v) , with ∇v the tensor derivative of the velocity
v. Both interpretations give the same result, independent of the coordinate system —provided
v.∇ is interpreted as
the covariant derivative. Turbulence is the time dependent chaotic behavior seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole: the culmination of time dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the Reynolds number quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.
DERIVATION OF THE NAVIER-STOKES EQUATIONS The derivation of the Navier–Stokes equations begins with an application of Newton’s second law: conservation of momentum (often alongside mass and energy conservation) being written for an arbitrary portion of the fluid. In an inertial frame of reference, the general form of the equations of fluid motion is:
where
v is the flow velocity, ρ is the fluid density, p is the pressure, Τ is the (deviatoric) stress tensor, and f
represents body forces (per unit volume) acting on the fluid and
∇ is the del operator.Vesely and Franz (2001).
This is a statement of the conservation of momentum in a fluid and it is an application of Newton's second law to a continuum; in fact this equation is applicable to any non-relativistic continuum and is known as the Cauchy momentum equation. This equation is often written using the material derivative Dv/Dt, making it more apparent that this is a statement of Newton's second law:
The left side of the equation describes acceleration, and may be composed of time dependent or convective effects (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of body forces (such as gravity) and divergence of stress (pressure and shear stress).
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Semi-Implicit Backward Euler Scheme for the Navier-Stokes 3d Problems: A Review
Numerical Solution of the Navier-Stokes Equations The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation. Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. Hairer, Ernst; Lubich, Christian; and Wanner, Gerhard (2003). To counter this, time-averaged equations such as the Reynolds-averaged Navier-Stokes equations (RANS), supplemented with turbulence models, are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Some models include the Spalart-Allmaras, k-ω (k-omega), k-ε (k-epsilon), and SST models which add a variety of additional equations to bring closure to the RANS equations. Vesely and Franz (2001). Other technique for solving numerically the Navier–Stokes equation is the Large eddy simulation (LES). This approach is computationally more expensive than the RANS method (in time and computer memory), but produces better results since the larger turbulent scales are explicitly resolved. Together with supplemental equations (for example, conservation of mass) and well formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations. The Navier–Stokes equations assume that the fluid being studied is a continuum not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations. Depending on the Knudsen number of the problem, statistical mechanics or possibly even molecular dynamics may be a more appropriate approach. Another limitation is very simply the complicated nature of the equations. Time tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations which are an area of current research. For this reason, these equations are usually written for Newtonian fluids. Studying such fluids is "simple" because the viscosity model ends up being linear; truly general models for the flow of other kinds of fluids (such as blood) do not, as of 2011, exist.
THE SEMI-IMPLICIT BACKWARD EULER METHOD FOR NAVIER-STOKES According to Guillen-Gonzalez (2010), Problem: Find u:
Ω × (0, T ) → R 3 and p : Ω × (0, T ) → R
such that :
∂u ∂ t + ( u .∇ ) u − ∆ u + ∇ p = f , ∇ .u = 0 u ( t ) |∂ Ω = 0 u |t = 0 = u 0
Mixed variational formulation of the problem (i): Find and v
q ∈ L20 (Ω)
in Ω × ( 0 , T ) in Ω × ( 0 , T ) on ∂ Ω × ( 0 , T ) in Ω
1 3 u (t ), p (t ) ∈ H 01 (Ω)3 × L20 such that v v ∈ H 0 (Ω)
(i )
18
Onanaye A. S. & Odekunle M. R.
( u ′ , v ) + ( ∇ u , ∇ v ) − ( p , ∇ .v ) + c ( u , u , v ) = ( f , v ) a .e . in ( 0 , T ) a .e . in ( 0 , T ) ( ∇ .u , q ) = 0 u (0 ) = u 0 in Ω
( ii )
Where
c (u , w, v ) = ((u .∇ ) w, v ) + 12 (∇.u , w.v ) = ...other reformulations Anti-symmetry : c (u , w, v ) = 0
for any
u ∈ H 01 (Ω)3
Existence Theorem The following Inf-Sup condition hold: there exists
Inf Sup q∈L20 ( Ω ) v∈H 01 ( Ω )2
∫
Ω
v
q∇.v q
H 1 (Ω )
β ≥ 0such that
≥ β ( Inf − Sup )
L2 ( Ω )
Moreover, the problem (ii) has (at least) a weak solution with
u ∈ L∞ (0, T ; L2 (Ω)) ∩ L2 (0, T ; H 1 (Ω)),
p ∈ H −1 (0, T ; L2 (Ω))
and the following energy inequality holds
u (t )
2 L
2
+
∫
t 0
2
u (s)
H
1
≤ u0
2 L
2
+
∫
t 0
2
f (s)
H
−1
, fo r a ll t ∈ [ 0 , T ]
The Scheme Finite-Elements (FE) in space and Finite-Difference (FD) in time.
τh regular partition of Ω , FE associated to
τ h : vh ⊂ H 01 (Ω)d and Wh ⊂ L20 (Ω)
such that :
vh ⊂ Pm , with m ≥ 1 and Wh ⊂ Pm −1 (vh , wh ) satisfying the discrete Inf-Sup condition (uniform with respect to h) Inicialization: Let
uh0 = PL2 ,V u0 h
Step n+1: known
U hn ∈ vh and Phn ∈ Wh , compute U hn +1 ∈ Vh and Phn +1 ∈Wh such that :
( i ii )
19
Semi-Implicit Backward Euler Scheme for the Navier-Stokes 3d Problems: A Review
Uhn+1 −Uhn ,Vh + (∇Uhn+1, ∇Vh ) + c(Uhn ,Uhn+1,Vh ) − (Phn+1, ∇.Vh ) = ( f n+1,Vh ) k for allVh ∈Vh (∇.U n+1, q ) = 0 for all qh ∈Wh h h Stability
u ≈ u
Notation:
L2
, u ≈ ∇u
L2
Theorem (Unconditional Stability): The following weak (a priori) estimates for the velocity (discrete version of (iii)):
U
2
r −1
2 2 h
∑
+ k
U
n +1 h
2
r −1
+
n=0
∑
U
n +1 h
n h
−U
≤ U
+
0
f
n=0
2 L2 ( 0 ,tr ; H
−1
)
(iv )
1 in k Pishn +bounded
Moreover, one has the following (weighted) strong estimates for the pressure: 4
I 3 L2 Weighted strong estimates for the pressure: by using the following bound for the convective terms in (in 3D n n +1 n n +1 n n +1 Vh1 domains): c(U h ,U h , Vh ) ≤ C ( U h 3 U h + U h U h 3 Vh L
L
Error Estimates for the Velocity Error in Velocity:
e n = U (tn ) − U hn = U (tn ) − I hU (tn ) + I hU (tn ) − U hn 144244 3 14 4244 3 e nj
Error in Pressure:
ehn
z n = P (tn ) − Phn = P (tn ) − J h P (tn ) + J h P (tn ) − Phn 144244 3 14 4244 3 z nj
I h : H 01 (Ω) → Vh , J h : L20 (Ω) → Wh
where
1.
zhn
are two interpolation operators,
I h discrete free-divergence interpolation, i.e. assuming tha satisfies: I h
(∇.(U − I hU ), qh ) = 0 for all qh ∈ Wh 2. Take ( I
h
, J
h
) ( U , P )as
the Stokes projector of ( U , P ) onto i.e. V h × W
h
(∇(U − I hU ), ∇Vh ) − ( P − J h P, ∇.Vh ) = 0 for allVh ∈ Vh ( I hU , J h P ) ∈ Vh × Wh (∇.(U − I hU ), qh ) = 0 for all qh ∈ Wh Hypothesis •
Approximation
I hU − U
H1
+ Jh P − P
L2
I hU − U
L2
+ Jh P − P
H −1
≤ Ch m ( U
H m+1
+ P
≤ Ch( I hU − U
H1
Hm
)
+ Jh P − P
L2
)
20
Onanaye A. S. & Odekunle M. R.
•
Stability
I hU
L∞ IW 1, 3
ε n+1 =
≤ C( U
H2
+ P
H1
) Consistence error in time:
U (tn +1 ) − U (tn ) − U t (tn +1 ) + (U (tn +1 ) − U (tn ).∇(tn +1 ) k
= 21k ∫
tn+1
tn
(tn − t )U tt (t )dt +
(∫
tn+1
tn
U t (t )dt.∇U (tn +1 )
)
:= ε1m +1 + ε 2m +1 U (t n +1 ) − U (t n ) − U t (t n +1 ) + (U (t n +1 ) − U (t n ).∇(t n +1 ) k t n+1 t n+1 = 21k ∫ (t n − t )U tt (t )dt + ∫ U t (t )dt.∇U (t n +1 ) tn tn m +1 m +1 := ε 1 + ε 2
ε n +1 =
Equations of the Error n +1 n e h − e h , V h + ∇ e hn + 1 , ∇ V h − ( z hn + 1 , ∇ .V h ) k n +1 = ε C O N S IS T E N C Y ,Vh n +1 n ej − ej ,Vh D IS C R E T E D E R IV A T IO N IN T IM E − k n +1 n +1 STO K ES − ∇ e j , ∇ V h + z j , ∇ .V h n n +1 − C kc U h , e ,Vh A N T IS Y S M M E T R Y n − C k c ( e , U ( t n + 1 ), V h ) N O N L IN E A R R E S T
(
(
)
) (
)
(
( ∇.e
n +1 h
)
)
, qh = 0 Discrete free − Div.
Error Estimates (Local): Taking as Test Functions
(Vh , qh ) = 2 k (ehn +1 , z hn +1 ) : 2
2
2
ehn +1 − ehn + ehn +1 − ehn + k ehn +1 ≤ Ck 2 ∫
t n +1
tn
(U
+ Ch 2( m +1) ∫
t n +1
tn
2 tt H −1
+ Ut
2 L2
)
2
CONSISTENCY
2
Ut
(
+ Ckh 2 m U (t n +1 )
H m +1 2 H m +1
(
DISCRETE FREE − DERIV . IN TIME + P (t n +1 )
2 Hm
) STOKES
)
+ Ckc U hn , ehn +1 , ehn +1 ( = 0 ) ANTISYM METRY + Ckh 2( m +1) U (t n +1 ) + Ck U (t n +1 )
2 ∞ I1,3
2 ∞ I 1,3
ehn
2
U (t n +1 )
2 H m +1
NON LINEAR REST
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Semi-Implicit Backward Euler Scheme for the Navier-Stokes 3d Problems: A Review
Error Estimates (Global) Adding in n=0,…,r-1, for any r=1,…, N, and using the Discrete Gronwall’s Lemma: r 2 h
e
r −1
+ k∑ e
2
r −1
n =0
(
2
+∑ e
n +1 h
n +1 h
n h
−e
n=0
2
≤ ec eh0 + Ck 2 ∫
tr
0
Where C depends on
(U
2 tt −1
+ Ut
U L∞ ( L∞ IW 1,3 )
2 0
) + Ch
2( m +1)
Ut
2 m +1
)
.Then, assuming a regulator enough exact (U , P ) : solution
U ∈ L∞ H m +1 , P ∈ L∞ H m , U t ∈ L2 H m +1 , U tt ∈ L2 H −1 the following error estimates for the discrete error holds:
ehn
I ∞ L∞ I I 2 H 1
≤ C (k + h m ).
Plugging the interpolation error, one has the same order for the total error:
en
I ∞ L∞ I I 2 H 1
≤ C ( k + h m ).
CONCLUSIONS Inclusion, the Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as multiphase flow driven by surface tension. Generally, application to specific problems begins with some flow assumptions and initial or boundary condition formulation; this may be followed by scale analysis to further simplify the problem. Semi-implicit Backward Euler method is found to be sure way out of complexity that normally associated with their solutions.
REFERENCES 1.
Acheson, D. J. (1990), Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing Science Series,Oxford University Press, ISBN 0198596790
2.
Batchelor, G.K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0521663962.
3.
Currie, I. G. (1974), Fundamental Mechanics of Fluids, McGraw-Hill, ISBN 0070150001.
4.
Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2003). "Geometric numerical integration illustrated by the Störmer/Verletmethod".ActaNumerica12:399–450.doi:10.1017/S0962492902000144. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=1 0.1.1.7.7106.
5.
Guillen-Gonzalez F. (2010), A Lecture Note on Backward Euler Semi-Implicit Scheme for the
Navier-Stokes
3D, DPTO. E.D.A.N., University of Seville, 41080 Seville. 6.
Landau, L. D.; Lifshitz, E. M. (1987), Fluid mechanics, Course of Theoretical Physics, 6 (2nd revised ed.), Pergamon Press, ISBN 0 08 033932 8, OCLC15017127
7.
Polyanin, A. D.; Kutepov, A. M.; Vyazmin, A. V.; Kazenin, D. A. (2002), Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, London, ISBN0-415-27237-8.
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Onanaye A. S. & Odekunle M. R.
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117.