CFD ICFD I Computational Fluid Dynamics I - UDCcaminos.udc.es/info/asignaturas/201/CFD...
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CFD IComputational Fluid Dynamics I
CFD ICFD I Computational Fluid Dynamics IPablo Rodríguez-Vellando Fernández-Carvajal
Hochschule Magdeburg-Stendal
Universidad de A CoruñaEscuela Técnica Superior de Ingenieros de Caminos Canales y Puertos
Hochschule Magdeburg StendalFachbereich Wasser und Kreislaufwirtschaft
Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos
CFD IComputational Fluid Dynamics I
• First term, MSc International Master in Water Engineering, 6 ECTS
• Lectures timetable:
• Grades: Attendance + Courseworks
• Lecturers
– Pablo Rguez-Vellando
– Héctor García Rábade
– Jaime Fe Marqués
CFD IComputational Fluid Dynamics I
Main Bibliography
• G Carey J Oden ‘Finite Elements’ Prentice-Hall 1984G. Carey, J. Oden, Finite Elements , Prentice-Hall,1984
• A. Chadwick, Hydraulics in Civil Engineering, Allen&Unwin, 1986
• J. Donea, ‘Finite Element Methods for Flow Problems’ Wiley, 2003y
• J. Ferziger, M. Peric, Computational methods for Fluid Dynamics
• P. Gresho, R Sani, ‘ Incompressible flow and the finite element method’, Wiley, 2000
• O. Pironneau, ‘Finite Element Methods for Fluids’, Wiley, 1989
• J. Puertas Agudo, Apuntes de Hidráulica de Canales, Nino, 2000
• Singiresu Rao, ‘The Finite Element Method in Engineering’, Elsevier 2005
• O. C. Zienkiewicz, R.L. Taylor, ‘The Finite Element Method. Vol 3, Fluid dynamics’, Mc
Graw HillGraw Hill
CFD I
0. Introduction to CFD. Revision of concepts (6h) 4. End user programmes (20h)
Computational Fluid Dynamics I
0. Introduction to CFD. Revision of concepts (6h)
1. Open channel flow. A revision
2. Saint-Venant equations
2. Introduction to CFD
3. Mathematical preliminaries
4. End user programmes (20h)1. MATLAB (8h)
2. HEC-RAS (4h)
3. SMS//RMA2 (8h)
1. Governing equations (6h)
1. Navier-Stokes
2. Potential, stream function, stokes flow
3. Shallow Water equations
4. Convection-diffusion eq
2. Finite elements and fluids hydrodynamics (24 h)
1. Finite elements and fluids
2. Variational and weighted residuals methods
3 Discretization3. Discretization
4. Potential flow
5. Stokes flow
6. Stable velocity-pressure pairs
7. Unsteady convective flow
8. Penalty methods
9. Shallow water equations
10. Stabilizing techniques
11. Flow in porous media
12. Conservative transport
13. Non-isothermal transport of reactives
3. Introduction to Finite Volumes (4h)
CFD IComputational Fluid Dynamics I
CFDCFDI 1 Introduction to CFDI 1. Introduction to CFD
CFD I
• In previous subjects we have regarded the Open Channel and Pipe flows
Computational Fluid Dynamics I
• In previous subjects we have regarded the Open Channel and Pipe flows• In the pipe flow the geometry is given and the unknowns are the pressure p(x,t)
and the velocity v(x,t). Some computational approaches have been regarded (e.g. EPANET)EPANET)
p
• In the open channel flow there is a hydrostatic distribution of pressures theIn the open channel flow there is a hydrostatic distribution of pressures, the unknowns are the shape (depth y(x,t)) and the velocity. Some computational approaches have been regarded (e.g. HEC-RAS)
p(z)zy zy
CFD IComputational Fluid Dynamics I
• As we can recall, the one dimensional flow in channels depends on space(x) and time (t) and can characterized as
Gradually varied
Unsteady
Open channel
Rapidly varied
Open channel Gradually varied flowFlow profiles (Curvas de remanso)
Non-uniformSt d fl
Uniform (i=I)
Steady flow Rapidly varied flowBroadcrested weir, hydraulic jump, sudden discharge variations
0 t/
Uniform (i=I) variations,…0/ x
CFD I
• The open channel flow takes place into natural channels and also in irrigation,
Computational Fluid Dynamics I
The open channel flow takes place into natural channels and also in irrigation, navegation, spillways, sewers, culverts and drainage ditches
• Prismatic channels are assumed (all the cross sections are equal)• Basic notation B• Basic notation
y(x,t)
B
y( , )
v(x,t) Ah
xPz
• Depth (y), Stage (h) height from datum, Area (A), Wetted perimeter (P), Surface width (B), Ground height from datum (z)
x
• Hydraulic radius (R), (R=A/P)• Hydraulic mean depth (Dm), (D=A/B)
CFD I
• In previous subjects you have regarded the Open Channel and Pipe flows
Computational Fluid Dynamics I
• In previous subjects you have regarded the Open Channel and Pipe flows• Saint-Venant equations allow for a resolution of the one dimensional flow• The continuity equation is given by the conservation of mass as
0
xv
BA
xyv
ty
• The dynamic equation is given by the conservation of momentum as
yvv
• In these differential equations the unknowns are the velocity v and the depth y for
0
iIg
xyg
xvv
tv
a given horizontal direction x• i is the geometric slope (i=-dz/dx) • I is the friction slope (I=- dE/dx)I is the friction slope (I dE/dx) • E is the Energy per unit weight given Bernoulli´s eq, E=z+y+v2/2g= z+pgv2/2g
CFD I
• Saint Venant equations assume :
Computational Fluid Dynamics I
• Saint-Venant equations assume :• The slope is small i<0.1• Flow straight and paralell. Hydrostatic distribution of pressures• Turbulent flow fully developedTurbulent flow fully developed• Uniform velocity within the section (Coriolis factor, =1)• Non-erodible boundaries• Prismatic channel
• Finding the value of dv/dx in the stationary continuity equation and substituting it in the stationary dynamic equation we obtain
IiIidy
• That can also be written as
gyvIi
gABvIi
dxdy
/1/1 22
That can also be written as yFryIiy 21
• Slow regime (Fr<1), fast regime (Fr>1)
CFD I
• The friction slope can be obtained from the Manning coefficient as
Computational Fluid Dynamics I
• The friction slope can be obtained from the Manning coefficient as2 2
4 3⁄
• The equation yFryIiy 21
has no analytic solution an has to be solve by a numerical method
Th l ti f hi h ill b i f th f
y
• The solution of which will be an expression of the form xyy M1
yn y M2
M3 xyc
CFD IComputational Fluid Dynamics I
CFD I
• The solution to the equation
Computational Fluid Dynamics I
yIi • The solution to the equation
can be solved on a finite element basis, to obtain
yFryy 21
xff kk
1
dx
xdf
wherekk x
IiFryx
**2
11
2/* FrFrFr 2/* III
xdx
where
and
21 /* kk FrFrFr 21 /* kk III
kkk gyBy
QFr 310
3422 2
/
/k
k ByByQnI
• The finite diference problem can be completed by using an intial condition
kk gyBy kBy
00 xyx
yn=y10 y0
y9
yn
x x10 x9 x8 x7 x6 x5 x4 x3 x2 x1 x0=0
CFD I
• First proposed in XIX c by Boudine (1861) and further developed by Bakhmeteff (1932)
Computational Fluid Dynamics I
• First proposed in XIX c. by Boudine (1861) and further developed by Bakhmeteff (1932)• Assuming that the initial condition is given downstream (y0 ) and that he stretch is long
enough for the normal depth to be reached, the iterative expression can be used N times varying the value of y from y0 up to yn at vertical equidistant intervals, and finally obtaining y g y y0 p y q , y gthe x for which the depth is the normal one
2009_10 Tramo 3k y fr I fr* I*
0 1 54 1 0023876 0 00364198 00 1,54 1,0023876 0,00364198 01 1,632 0,9188329 0,00309299 0,9606102 0,003367485 -3,001066462 1,724 0,8462737 0,00265301 0,88255329 0,002873 -13,86127333 1,816 0,7827859 0,00229591 0,8145298 0,002474459 -34,86001414 1,908 0,7268574 0,00200275 0,75482162 0,002149326 -69,29974365 2 0,6772855 0,0017596 0,70207141 0,001881175 -122,2435925 2 0,6772855 0,0017596 0,70207141 0,001881175 122,2435926 2,092 0,6331028 0,00155607 0,65519413 0,001657836 -202,0602687 2,184 0,5935233 0,00138424 0,61331306 0,001470155 -324,1347348 2,276 0,5579026 0,00123806 0,57571295 0,001311153 -521,809089 2,368 0,5257075 0,00111282 0,54180505 0,001175445 -892,257412
10 2,46 0,4964941 0,00100482 0,51110081 0,001058825 -2047,68149
1 71,92,12,32,52,7
do
Tramo 4
T
0,70,91,11,31,51,7
-2500 -2000 -1500 -1000 -500 0
Cal
ad
Distancia
Tra…
CFD I
• Even the one dimensional Saint Venant equations are difficult to resolve and
Computational Fluid Dynamics I
• Even the one-dimensional Saint-Venant equations are difficult to resolve and some numerical procedure is to be needed. Step, characteristics, finite differences, finite volumes and finite elements are some of those
• The extension of the Saint Venant equations to the three dimensions are called• The extension of the Saint-Venant equations to the three dimensions are called the Navier-Stokes. They are also made up of continuity and dynamic equation
0
wvu
zyx
2
2
2
2
2
21zu
yu
xu
xpf
zuw
yuv
xuu
tu
x
2
2
2
2
2
21zv
yv
xv
ypf
zvw
yvv
xvu
tv
y
2221 wwwpwwww
the unknowns in these equations will be the velocities u(t,x,y,z), v(t,x,y,z), w(t,x,y,z)
222
1zw
yw
xw
zpf
zww
ywv
xwu
tw
z
and the presure p(t,x,y,z)
CFD I
• That is
Computational Fluid Dynamics I
That is
0
zw
yv
xu
0iiu zyx
xfzu
yu
xu
xp
zuw
yuv
xuu
tu
2
2
2
2
2
21
2221 1
0iiu ,
yfzv
yv
xv
yp
zvw
yvv
xvu
tv
2
2
2
2
2
21
zfwwwpwwwvwutw
2
2
2
2
2
21
jjiiijijti upfuuu ,,,,
1
with boundary conditions: Dirichlet in (prescribed velocity)
zyxzzyxt
222
ii bu t bou da y co d t o s c et (p esc bed e oc ty)Newman in 2 (prescribed normal stress )
with initial conditions (unsteady flow)
ii buijij tn
jiji xuxu 00 ,
CFD IComputational Fluid Dynamics I
• Anyway, in most of the cases some others equations are used to provide a
simplified b t meaningf l sol tion to the flo problems Among these e cansimplified but meaningful solution to the flow problems. Among these we can
quote as some of the most important
–Potential and stream function equations
–Stokes flow eqs.Stokes flow eqs.
–Shallow water flow eqs. (SSWW)
CFD IComputational Fluid Dynamics I
• The potential flow equation is a simplification that uses the potential variable to
solve the continuity equation
• In the stream/vorticity formulation the u and p variables are written in terms of
the variables and , obtaining in this way simplified N-S equations
• In the Stokes equations the convective term is dropped
• The Shallow Water equations are the result of the integration in depth of the
three dimensional equations, therefore a two dimensional model is obtained
CFD I
• The flow in a porous media simplifies Navier-Stokes eq and is also to be
Computational Fluid Dynamics I
• The flow in a porous media simplifies Navier-Stokes eq. and is also to be
considered
• Once the velocity field is obtained, we can use it as an input value to resolve the
transport equation that gives the concentration of a given species in the flowt a spo t equat o t at g es t e co ce t at o o a g e spec es t e o
• The transport equation can be also considered for non-isothermal reactivesThe transport equation can be also considered for non isothermal reactives
• The equations of the transport of sediments are also needed for the case inThe equations of the transport of sediments are also needed for the case in
which non-soluble substances are included in the flow
• For convective enough flows, a turbulent model is to be required
CFD I
• With respect to the dynamic macroscopic behaviour, flows can be regarded
Computational Fluid Dynamics I
With respect to the dynamic macroscopic behaviour, flows can be regardedas laminar or turbulent
• The laminar flow is ordered and it takes place in layers• The laminar flow is ordered and it takes place in layers
• In the turbulent flow particles move on an irregular fluctuant and erraticIn the turbulent flow, particles move on an irregular, fluctuant and erraticway -> turbulents models are required
• This situation takes place for a Reynolds number Re(=UL/ > 2000
• The Reynolds number indicates the weight of the convection with respect tothe viscous losses
CFD IComputational Fluid Dynamics I
• When the Reynolds number is large enough, the velocity unknown is split
into a mean velocity U and a fluctuating term that depends on time u’(t),
leading to u(t)=U+u’(t)
• The most common models are the algebraic, de one equation models
(Prandtl's Baldwin-Barth etc ) and the two eq (k k )(Prandtl s, Baldwin-Barth, etc...) and the two eq. (k k ,...)
CFD IComputational Fluid Dynamics I
• The FEM was developed in the 50s to be applied to the aeronauticengineering
• Advantages:• Advantages:– Suitable to model complex geometries
– Consistent treatment of b.c.
– Possibility of being programmed in a flexible and general way
• Fluid materials change their shape and that leads to a importantcomplexity
• Structural or heat problems lead to a diffusive equation that turns into affsymetric stiffness matrices
• For those cases, Galerkin formulation leads to convergent iterativesolutions in an easy waysolutions in an easy way
CFD IComputational Fluid Dynamics I
• The presence of a convective acceleration in the fluids formulation leads to theobtaining of non-symmetric stiffness matrices
• That is the reason of the Galerkin formulation not being appropriate anymore.When using it, spurious wiggles show up in the solutiong p gg p
CFD IComputational Fluid Dynamics I
• In order to do avoid these oscillations, some techniques have been developed since the
70s which are known as stabilization techniques. The most important of which are
SUPG (Streamline Upwind Petrov Galerkin)– SUPG (Streamline Upwind Petrov-Galerkin)
– GLS (Galerkin Least Squares)
– FIC (Finite Increment Calculus),...
• A correct coupling in the selection of the pressure and velocity variables is required for
convergenceg
• The heterogeneity of the unknowns require the use of the so-called mixed and penalized
methodsmethods
• The mesh refinement also leads to the stabilization (but means high computational costs
index
0. Introduction to CFD (4 h)
Computational Fluid Dynamics I
0. Introduction to CFD (4 h)
1. Governing equations (6 h)
1. Navier-Stokes
2. Potential, stream function, stokes flow
3. Shallow Water equations
4. Convection-diffusion eq
2. Finite elements and fluids hydrodynamics (26 h)
1. Finite elements and fluids
2. Variational and weighted residuals methods
3. Discretization
4. Potential flow
5. Stokes flow
6. Stable velocity-pressure pairs
7 Unsteady convective flow7. Unsteady convective flow
8. Penalty methods
9. Shallow water equations
10. Stabilizing techniques
3. Flow in porous media (6 h)
4. Conservative transport (6 h)
• Non-isothermal transport of reactives
• Transport of sediments
• Turbulence models
• Finite volumes
introduction to CFDderivative operators
f ( ) i 1D l fi ld
derivative operatorscomputational fluid dynamics I
• f (x,t) is a 1D scalar field
• f (x,t) is a 3D vectorial field
• · = scalar product332211 babababa jj a·b
• Index notationj
iji x
uu
,
• Gradient , divergence
zyx
,,
• Laplacian
y
2
2
2
2
2
2
zyx,,
zyx
introduction to CFDReference SystemReference Systemcomputational fluid dynamics I
• Lagrangian coordinates (the net follows the particle)– Not able to model big deflections (even in structures)Not able to model big deflections (even in structures)– Allows to follow the interface between different materials
• Eulerian coordinates (the net is fixed and the fluid moveswith respect to it)
– Allows for a characterization of big deflections (fluids)– Difficulties to evaluate interfaces and free surfaces
• ALE coordinates (mixture of both)– The net moves with an independent velocity from that of the
ti lparticles
introduction to CFDeulerian coordinateseulerian coordinatescomputational fluid dynamics I
• In the Lagrangian coordinates there are no convective efects and the materialderivative is just a temporal derivative
I th E l i di t th i l ti t f th t i l• In the Eulerian coordinates there is a relative movement of the materialcoordinates with respect to the spatial ones, and the material derivative of anscalar field f is given by
xffddf j
ftf
dtdf
·utxtdt j
ftdt
ffdfdfdffdf )(
jj
i
xfu
tf
dtdz
zf
dtdy
yf
dtdx
xf
tf
dttxdf
),(
introduction to CFDeulerian coordinateseulerian coordinatescomputational fluid dynamics I
• The total derivative of a vectorial field is given by
xffdf j ff dt
xxf
tf
dtdf j
j
iii
fuff
·tdt
d
jj
i
xfu
tf
dtdz
zf
dtdy
yf
dtdx
xf
tf
dttxdf
1111111 ),(
jj
i
xfu
tf
dtdz
zf
dtdy
yf
dtdx
xf
tf
dttxdf
2222222 ),(
j
jj
i
xfu
tf
dtdz
zf
dtdy
yf
dtdx
xf
tf
dttxdf
3333333 ),(
jxtdtzdtydtxtdt
introduction to CFDeulerian coordinateseulerian coordinatescomputational fluid dynamics I
• The compact integral forms are:
da v:u)vu,(
dqqb v·)u,(
dvuu,va ·)(
dqwqw ),(
dc u)·v·(w)u,w;v(
duwuwc )·v(),;v(
dhh)(where
j
iij x
u
u
ii vu uu
dhwhwN
N),(
j
i
j
i
xx v:u
iji x
uvw
u·v·w
i
i xu
ii xv
xuvu
·jxii
governing equations
computational fluid dynamics I
CFDCFDI2 Governing EquationsI2. Governing Equations
governing equationsstress() and strain() of fluids
• For solids Hookes´s law states
stress() and strain() of fluidscomputational fluid dynamics I
E• For solids, Hookes s law states• For Newtonian fluids (air and water are included) Newton´s viscosity law
states
E
du
where is the dynamic viscositydn
smkg·
and is the cinematic viscosity
sm2
• For no-newtonian fluids (plastics, coloidal suspensions, emulsions,...) theviscosity is not a constant
• For the non-frictional flow or non-viscous flow (inviscid) viscosity isnegligiblenegligible
• In what follows, the Navier-Stokes eq., governing the viscous flow, aredescribed for compressible fluids (gases is not a constant) and fornon-compressible fluids (liquids, c)
governing equationscontinuity equation
• The principle of conservation of mass states that in any time interval and for any
continuity equationcomputational fluid dynamics I
• The principle of conservation of mass states that in any time interval and for any control volume the volume of mass entering must equal the volume of mass leaving, i.e.
outoutinin QQ outoutinin QQ
outoutoutininin AuAu
• As velocity and density depend on time and space, the equilibrium of mass in a differential volume dxdydz can be stated from
dydzdxux
u
udydzy
xz
governing equationscontinuity equation
• The flux of mass per second this is is equal to (subtract in figure)
continuity equationcomputational fluid dynamics I
dxdydz• The flux of mass per second, this is , is equal to (subtract in figure) dxdydzt
dxdydzwz
dxdydzvy
dxdydzux
dxdydzt
• Since the control volume is independent of time
y
F i ibl fl id i t t d th ti it ti lt i t
wz
vy
uxt
• For incompressible fluids is a constant and the continuity equation results into
0
iiuwvu u· iizyx ,
governing equationsdynamic equation
• Newton´s second law states that
dynamic equationcomputational fluid dynamics I
• Newton s second law states that mav
dtdm
dtmvdF
• In the control volume there is no variation in mass, and therefore
ii dxdydzadF • The equilibrium of forces gives
ii y
dxdzdyyx
dydzdxxxxx
y dydzxx
dxdyzx
dxdzdyyyx
yxxx
xz
y
dxdydzz
zxzx
dxdzyxz
governing equationsdynamic equation
• Newton´s second law can be written for the x direction as
dynamic equationcomputational fluid dynamics I
• Newton s second law can be written for the x direction as
dydzdxx
dydzdxdydzBdF xxxxxxxx
dxdzdyx
dxdz yxyxyx
where Bx are the body forces in the x directionDi idi b th t l l d ki th ti f th th
dxdydzx
dxdy zxzxzx
• Dividing by the control volume and making the same operations for the three dimensions in space it is obtained
Ba zxyxxxxx
zyxxx
zyxBa zyyyxy
yy
zyx
Ba zzyzxzzz
governing equationsstresses in solids
• Which is the value of ? Let us see first how solids behave
stresses in solidscomputational fluid dynamics I
• Which is the value of ij ? Let us see first how solids behave
• In solids the strains are related to the stresses asIn solids the strains are related to the stresses as
,...zzyyxxxx E
1
where E is the Young modulus, is the Poisson ratio and G is the Modulus of
,...G
xyxy
Rigidity or shear modulus
governing equationsstresses in solids
• The volume dilation e can be defined as follows
stresses in solidscomputational fluid dynamics I
• The volume dilation e can be defined as follows
d d d
dxdydzdxdydzVVe xxxxxx
111dxdydzV
32121e zzyyxxzzyyxx
where is the mean of the three normal stressesTh fi t t i th f b d
EE zzyyxxzzyyxx
• The first strain can therefore be expressed as
xxxxxxzzyyxxxxzzyyxxxx 3111 xxxxxxzzyyxxxxzzyyxxxx EEE
13
1 xxxxE
governing equationsstresses in solidsstresses in solidscomputational fluid dynamics I
• Therefore, writing in terms of e
3 EeEE
• Noting that Young´s and shear modulus and Poisson´s ratio are related as
211111
xxxxxx
Noting that Young s and shear modulus and Poisson s ratio are related as
12EG
• It is obtained
eGG xxxx 22 eG xxxx
21
governing equationsstresses in solids
• Subtracting from both sides of the former equation we obtain
stresses in solidscomputational fluid dynamics I
• Subtracting from both sides of the former equation we obtain
eEGGeGG xxxxxx
2132122
2122
eGGeGGeGGG xxxxxxxx
31
32
2122
31
2122
21312
2122
• Or
Si il l
32 eG xxxx
2 eG• Similarly
3
2G yyyy
32 eG zzzz
• From the first equations it is already known that 3
xyxy G
G yzyz G
zxzx G
governing equationsstresses in fluidsstresses in fluidscomputational fluid dynamics I
• Up to this point we have been concerned with solids It has been shownUp to this point we have been concerned with solids. It has been shown empirically that stresses in fluids are related not to strain but to time rate of strain
• We have just shown that
32 eG xxxx
• Replacing the rigidity modulus by a quantity in terms of its dimensions (F/L2), the stresses in fluids would be of the form
3xxxx
3
2 2 et
LFT xxxx /
• Where the proportionality constant is known as the dynamic viscosity and has the dimensions (FT/L2)=(M/TL)
• The equations result into• The equations result into
,...te
txx
xx
322 ,...
txy
xy
t
governing equationsstresses in fluidsstresses in fluidscomputational fluid dynamics I
• Taking the mean pressure as –p, the equations are
te
tp xx
xx
322 xyxy
e 2te
tp yy
yy
322
ezz 22
yzyz
L t fi d t th l f th ti d i ti f d i t f
ttp zz
zz
32 zxzx
• Let us now find out the value of the time derivatives of xy and e in terms of u,v and w
governing equationsstresses in fluidsstresses in fluidscomputational fluid dynamics I
• If the coordinates of a point before deformation are x,y,z and after deformations are xy+, z+the strains are given by
xxx
yyy
zzz
• The rate of strain and volume dilation would be therefore
xyxy
yzyz
zxzx
,...xu
txxttxx
wvue
,...xv
yu
txtyxyttxy
u·
zyxtt zzyyxx
governing equationsdynamic equation
• And consequently the stresses result into
dynamic equationcomputational fluid dynamics I
• And consequently the stresses result into
u·
322
322
xup
te
tp xx
xx
yu
xv
txy
xy
u·
322
yvpyy
2
zv
yw
yz
It i bt i d f th fi t di i
u·
322
zwpzz
xw
zu
zx
• It is obtained for the first dimension
zyxB
zuw
yuv
xuu
tua zxyxxx
xx
111
zyxzyxt
xw
zu
zyu
xv
yxup
xB
zuw
yuv
xuu
tu
x
1121 yyy
governing equationsdynamic equation
• The first dynamic equation is transformed into
dynamic equationcomputational fluid dynamics I
• The first dynamic equation is transformed into
xw
zu
zyu
xv
yxup
xB
zuw
yuv
xuu
tu
x
1121
xzw
zu
yu
xyv
xu
xpB
zuw
yuv
xuu
tu
x
2
2
2
2
22
2
2
21
wvuuuupBuuuu 2221
zyxxzyxx
pBz
wy
vx
ut x
222
2
2
2
2
2
21 uuupBuwuvuuu
• Proceeding in the same way for for y and z, the 3D Navier equations are finally
1
222 zyxx
Bz
wy
vx
ut x
jjiiijijti upfuuu ,,,,
1
1 fuuuu
p
t1·
governing equationsdynamic equation
• That is
dynamic equationcomputational fluid dynamics I
That is
0
zw
yv
xu
xfzu
yu
xu
xp
zuw
yuv
xuu
tu
2
2
2
2
2
21
222
yfzv
yv
xv
yp
zvw
yvv
xvu
tv
2
2
2
2
2
21
wwwpwwww
2221
with boundary conditions: Dirichlet in (prescribed velocity)
zfzw
yw
xw
zp
zww
ywv
xwu
tw
222
1
ii bu t bou da y co d t o s c et (p esc bed e oc ty)Newman in 2 (prescribed normal stress )
with initial conditions (unsteady flow)
ii buijij tn
jiji xuxu 00 ,
• When the flow is non-isothermal, the temperature of the fluid has to be solved making use of the energy equation, which represents the conservation of energy
governing equationsstokes flow
• The Stokes flow simplification is obtained when the flow is taken as steady and
stokes flowcomputational fluid dynamics I
• The Stokes flow simplification is obtained when the flow is taken as steady and the convective term is dropped. For the two dimensional case leads to
0 vu
01
xfuxp
0
yx
x
01
yfvyp
• The equation can be solved in terms of the variables as – Stream function formulation– Stream-function-vorticity formulation– Velocity presure
governing equationspotential flowpotential flowcomputational fluid dynamics I
• A flow is said to be inviscid (or non-viscous) when the effect of viscosity is small compared to the other forces (convection)
• This can be assumed for instance in flow through orifices, over weirs or in channelsg• A flow is said to be irrotational when its particles do not rotate and maintain the same
orientation wherever along thr streamline
irrotational rotational
governing equationspotential flowpotential flowcomputational fluid dynamics I
• In irrotational flows the rotational of the velocity vector is zero
kji
0kji
kji
urot
yu
xv
xw
zu
zv
yw
wvuzyx
• Therefore in rotational flows it is verified that
wvu
0
zv
yw
0
xw
zu 0
yu
xv
• Far from the boundaries, most of the flows of fluids with low viscosity (such as air and water) behave as irrotational and these simplification can be assumed, that is why the
y y
inviscid flow can be considered in certain occasions as irrotational
governing equationspotential flow
• The potential flow equations are a simplified version of the N-S equations in which the
potential flowcomputational fluid dynamics I
• The potential flow equations are a simplified version of the N-S equations in which the potential function is used to solve the continuity equation
• We define in such a way that its partial derivatives with respect to the space, give the velocity in that directiony
• Substituting this expression into the 2-D continuity equation it is obtained
ux
v
yv
g p y q
0
yv
xu
• It is also verified that
02
2
2
22
yx
It is also verified that
and the assumption of a velocity potential requires the flow to be irrotational
xv
yu
yxxy
and the assumption of a velocity potential requires the flow to be irrotational
governing equationspotential flow
• With this formulation we can solve problems such as flow around a cylinder flow out of an
potential flow computational fluid dynamics I
With this formulation we can solve problems such as flow around a cylinder, flow out of an orifice or around an airfoil
• The flow through a saturated homogeneous porous media results as well in a Laplacian, as the Darcy´s law is given by , where h is the water level, can be dxdhku written as
ku
where k is the hydraulic conductivity• Taking this equation to the continuity equation it is obtained
assuming k as a constant
fkk ·
g
governing equationspotential flowpotential flowcomputational fluid dynamics 1
• The governing equations of the two dimensional potential flow are therefore given by
22
in 02
2
2
22
yx
where the velocity components are given by
with the boundary conditions
xu
yv
with the boundary conditionsDirichlet in
Newman in 2
0
0VllV yxn
2
were lx and ly are the direction cosines of the outward unit vector n to 2
0yxn yxn
governing equationsstream functionstream function computational fluid dynamics 1
• The stream function ( formulation is an alternative way of describing the motion ofthe fluid that has some important advantages compared to the velocity-pressureformulationformulation
• The streamline (línea de corriente) is a line that connects points at a given instantwhose velocity vectors are tangent to the line
• The path line (línea de trayectoria) connects points through which a fluid particle offixed identity passes as it moves in space
I t d fl b th li th• In steady flow both lines are the same
• Since the velocity vector meets the streamlines tangentially no fluid can cross thestreamline
• In the stream-function formulation the unknown is defined as
u v
y x
governing equationsstream function
• If a unit thickness of the fluid is considered is defined as the volume rate
stream function computational fluid dynamics I
• If a unit thickness of the fluid is considered, is defined as the volume rate (vol per unit distance/T) of fluid between streamlines AB and CD. Let C’D’ be a streamline very closed to CD. Let the flow between CD and C’D’ be d
D’
C’
Ddx
dyv
u D
PC
C
B
• At a point P (with velocities u and v), the distance between CD and C´D´ is denoted by –dx and dy
A
denoted by –dx and dy• Since no fluid crosses the streamlines, the volume rate of flow across dy is u and
the volume rate across –dx is v, therefore
ddd vdxudyd
governing equationsstream functionstream function computational fluid dynamics I
• Therefore
A d th ti it ti i t ti ll ti fi d b th t f ti
uy
v
x
• And the continuity equation is automatically satisfied by the stream function
0
xyyxyv
xu
• If the flow is irrotational, the equation to be satisfied is
yyy
0
uv
• Substituting u and v by its values in terms of it is obtained yx
0
u
• And therefore
0
xyxx
222 0222
yx
governing equationsshallow waters
• The equations governing the steady 2 D Newtonian flow are
shallow waterscomputational fluid dynamics I
• The equations governing the steady 2-D Newtonian flow are
0
yv
xu
xfuxp
yuv
xuu
tu
1
or identically
yfvyp
yvv
xvu
tv
1
0 fu
1or identically
• But this is just a theoretical example in which the flow is assumed to have
0, iiu ufuu
pt
· 2,1i
j pnull thickness
• If we want to make a more adequate approach that takes into account the third dimension we have to use the Shallow Water equations (SSWW)third dimension we have to use the Shallow Water equations (SSWW)
governing equationsshallow waters
• The equations governing the steady 2 D Newtonian flow are
shallow waterscomputational fluid dynamics I
• The equations governing the steady 2-D Newtonian flow are
0
yv
xu
xfuxp
yuv
xuu
tu
1
or identically
yfvyp
yvv
xvu
tv
1
0 fu
1or identically
• But this is just a theoretical example in which the flow is assumed to have
0, iiu ufuu
pt
· 2,1i
j pnull thickness
• If we want to make a more adequate approach that takes into account the third dimension we have to use the Shallow Water equations (SSWW)third dimension we have to use the Shallow Water equations (SSWW)
governing equationsshallow watersshallow waterscomputational fluid dynamics I
•The assumptions to be made are•The assumptions to be made are
– The distribution of the horizontal velocity along the vertical direction is assumed y gto be uniform
An integration in height is carried out and the horizontal velocity is taken as the– An integration in height is carried out, and the horizontal velocity is taken as the mean value of the horizontal velocities along the vertical direction
– The main direction of the flow is the horizontal one, and only very small flows take place on vertical planes
– The acceleration in the vertical direction is negligible compared to gravity and a hydrostatic distribution of the pressure is assumed
governing equationsshallow waters. continuity eq.
• Integrating the continuity equation along the z axis
shallow waters. continuity eq.computational fluid dynamics I
• Integrating the continuity equation along the z-axis
0
wvu
h
0 hh
hwhwdvdu
0
zyx h
hbH=h+hb
A th L ib i l t b i th d i ti i t th i t l i i
0 bhh
hwhwdzy
dzx
bb
• As the Leibniz rule to bring the derivatives into the integral sign gives
0 xhhu
xhhuudz
xdz
xu b
b
h
h
h
h
it is obtainedxxxx hh bb
0 bb
b
hb
b
h
hwhwhhvhhvvdzhhuhhuudz
bb
hb
h yyyxxxbb
governing equationsshallow waters. continuity eq.
• w(h) (vertical component of the velocity on the surface) is given by
shallow waters. continuity eq.computational fluid dynamics I
• w(h), (vertical component of the velocity on the surface) is given by
hvyhhu
xh
th
dtdhhw
• Substituting in the former equation
hh bhh
• Noting that and taking and renaming the main velocities as
0
th
thvdz
yudz
xb
hh bb
0hb• Noting that , and taking and renaming the main velocities as0t
b
uudzH
uh
hb
1
vvdzH
vh
hb
1
the continuity equation is obtained asb
0
vHuHh 0
yxt
governing equationsshallow waters. dynamic eq.
• As the vertical acceleration is negligible the third dynamic equation
shallow waters. dynamic eq.computational fluid dynamics I
As the vertical acceleration is negligible, the third dynamic equation
zfzw
yw
xw
zp
zww
ywv
xwu
tw
2
2
2
2
2
21
can be written as01
zfzp
yy
• Integrating this equation in depth and assuming the atmospheric pressure to be zero it is obtained
z
phh
dz
zpdzf
h
h
h
h zbb
phphphhf bbz
• Deriving with respect to x and y
phf 1 phf 1xp
xfz
y
py
fz
governing equationsshallow waters. dynamic eq.
• The first dynamic equation results into
shallow waters. dynamic eq.computational fluid dynamics I
• The first dynamic equation results into
2
2
2
2
2
2
zu
yu
xu
xhff
zuw
yuv
xuu
tu
zx
• Adding the continuity equation multiplied by u, it is obtained
zyxxzyxt
2
2
2
2
2
2
zu
yu
xu
xhff
zw
yv
xuu
zuw
yuv
xuu
tu
zx
this is
2
2
2
2
2
22
zu
yu
xu
xhff
zuw
yuv
xu
tu
zx
as zyxxzyxt
wuwuvuvuuuuuwuvuu
22
zzyyxtzyxt
governing equationsshallow waters. dynamic eq.
• Integrating in depth the former expression
2222 uuuhuwuvuu
shallow waters. dynamic eq.computational fluid dynamics I
• Integrating in depth the former expression
xhhu
xhhudzu
xthhuudz
tb
b
h
h
h
h bb
222
222 zu
yu
xu
xhff
zuw
yuv
xu
tu
zx
xxxtt bb
dzHxhffhwhuhwhu
yhhvhu
yhhvhuuvdz
yh
hzxbb
h
hb
bbbb
u
• Taking into account that , it is obtained
yyy
hvyhhu
xh
th
dtdhhw
dHhffhhhhhhhhhhhhhhhhhhdhbbbh b
xhhu
xhhudzu
xthhuudz
tb
b
h
h
h
h bb
222
• Cancelling terms
dzHx
ffhvy
huxt
huhvy
huxt
huy
hvhuy
hvhuuvdzy hzxb
bb
bbbh
bbb
bb
u
dzHxhffuvdz
ydzu
xudz
th
hzx
h
h
h
h
h
h bbbb
u2
governing equationsshallow waters. dynamic eq.
• Taking mean velocities it is obtained
shallow waters. dynamic eq.computational fluid dynamics I
Taking mean velocities it is obtained
dzHxhff
yuvH
xHu
tuH h
hzxb
u
2
• The viscosity effects can be evaluated as
bs
h
hHuudz
2
2
2
2
u
where v is the turbulent viscosity• Where are the shear stresses acting on the surface (due to the wind action)
d th b tt (d t th h f th h l)
xxb
bsh yx
22
xx bs ,and on the bottom (due to the roughness of the channel)
iaws
WWCi 34
2
h
ib H
uVgnH
i
= Wind drag coefficient = Manning coefficient= Wind velocity components
h
WCn
iW y p= Air density
i
a
governing equationsshallow waters. dynamic eq.
• Developing the derivatives in the left hand side
shallow waters. dynamic eq.computational fluid dynamics I
• Developing the derivatives in the left hand side…
yHuvH
yvuv
yu
xHuH
xuu
tHuH
tu
yuvH
xHu
tuH
)(2 2
2
• Taking into account the continuity eq….
yyyxxttyxt
0
yHvH
yv
xHuH
xu
th
yvH
xuH
th
the former eq becames
uHvHuHuuuvHHuuH 2
vHyu
yHvH
yv
xHuH
xu
tHuH
xuuH
tu
yuvH
xHu
tuH
)(
uuuuvHHuuH 2
vHyuH
xuuH
tu
yuvH
xHu
tuH
0
governing equationsshallow waters. dynamic eq.
• The derivatives of the depth with respect to x and y are
shallow waters. dynamic eq.computational fluid dynamics I
• The derivatives of the depth with respect to x and y are
xh
xhh
xH b
• Carrying out the same operations for the y dimension, and developing the d i ti t ki i t t th l t i it i bt i d
xxx
derivatives taking into account the last expression it is obtained
34
2
2
2
2
2xaw
c HuVgn
HWWC
yu
xuvf
xhg
yuv
xuu
tu
hHHyxxyxt
34
2
2
2
2
2yaw
c HvVgn
HWWC
yv
xvuf
yhg
yvv
xvu
tv
where fc is the Coriolis factor
hHHyxyyxt
governing equationsshallow waters
• The shallow water equations result into
shallow waterscomputational fluid dynamics I
• The shallow water equations result into
0
yvH
xuH
th
34
2
2
2
2
2
h
xawc H
uVgnH
WWCyu
xuvf
xhg
yuv
xuu
tu
yxt
hyy
34
2
2
2
2
2
h
yawc H
vVgnH
WWCyv
xvuf
yhg
yvv
xvu
tv
with boundary conditionsimpermeability , (no slip)0u 0uimpermeability , (no slip)discharge contour stresses ,
t l l
0Nu 0Tu
QdsHuN
0NN 0TT
water level thth 0
governing equationsconvection-diffusion equation
• If in the N S dynamic equation we substitute the non linear velocities by a known
convection diffusion equationcomputational fluid dynamics I
• If in the N-S dynamic equation we substitute the non-linear velocities by a known velocity field and the rest of the velocities by the a scalar unknown we arrive to the convection diffusion equation that rules the transport of substances by convective and diffusive actionsconvective and diffusive actions.
• The equations are
fWVU
222 f
zyxzW
yV
xU
t
222
0 QkU
or in 1D
0 QkU jjjjt ,,,
0
QkU
where is the quantity being transported, k is the diffusion coefficient, Ui is the
0
Qx
kxx
Ut
known velocity field, and Q are the external sources of the quantity. These are also known as the Transport Equations
finite elements in fluids
computational fluid dynamics I
CFDCFDI 3 Finite Elements in FluidsI 3. Finite Elements in Fluids
finite elements in fluidsgeneral issuesgeneral issues computational fluid dynamics I
• There is no analytical solution for most engineering problems such asfluid flow
• The determination of the velocity and pressure field is required in a• The determination of the velocity and pressure field is required in adomain of infinite degrees of freedom
• The Finite Element Method (developed about 1950 for structures)The Finite Element Method (developed about 1950 for structures)substitutes the domain by another with a finite number of freedomdegrees, thus an approximation of the solution is obtained
• Some important names in the finite element history are Courant, Turner,Clough, Zienkiewicz, Brookes, Hughes,…
• Now it is used not only in structural mechanics but also in heatconduction, seepage flow, electric and magnetic fields, and of coursein fluid dynamicsin fluid dynamics
finite elements in fluidsgeneral issuesgeneral issues computational fluid dynamics I
sms.avi
finite elements in fluidsgeneral issuesgeneral issues computational fluid dynamics I
largo modulos.avi
2D h zoom at the mine.avi
3D H(x,y), water depth colour (only 600 days).avi
finite elements in fluidsgeneral issuesgeneral issues computational fluid dynamics I
200
250VEL
1.781251.66251.543751.4251.306251 1875
Y
100
150
1.18751.068750.9500020.8312520.7125010.5937510.4750010.3562510.2375010.118751.69975E-069.60324E-07
50
100 4.19696E-078.11084E-08
X0 100 200
0
2D H (water level).avi
finite elements in fluidsgeneral issuesgeneral issues computational fluid dynamics I
finite elements in fluidsgeneral issuesgeneral issues computational fluid dynamics I
The main way of solving continuum problems in the finite element method are the following
•The direct approach (matrix analysis), by using a direct physical reasoning to establish
the element properties. Requires very simple basic elements (bars, pipelines,…)
•Variational approach (e.g. Rayleigh-Ritz based method), in this method the stiffness
matrix is obtained as a result of the resolution of a variational problemmatrix is obtained as a result of the resolution of a variational problem
•Weighted residual approach (e.g. Galerkin Method), as a result of weighting the
differential equations and integrating them in the domain
finite elements in fluidsgeneral issuesgeneral issues computational fluid dynamics I
• Main steps of the finite element method– Subdivide the domain in a finite number of elements interconnected a the nodes,
where the unknowns (p u) are going to be determinedwhere the unknowns (p, u) are going to be determined
– It is assumed that the variation of the unknowns can be approximated by a simplefunction
– The approximation functions are defined in terms of the values of the fieldvariables at the nodes
– When the equilibrium or variational equations has been obtained the new finiteWhen the equilibrium or variational equations has been obtained the new finiteunknowns are introduced into the equations
– The system of equations is solved and the unknowns are determined at the nodes
– The approximation functions give the solution in the rest of the domain points
• Following, the fem solution of the one simple 1-D problem is to be considered6 t b ion a 6-step basis
finite elements in fluidsgeneral issuesgeneral issues computational fluid dynamics I
'...as the nature of the universe is the most perfect and the work of the Creator is wiser, there's nothing that takes place in the universe in which the ratio of maximum and minimum does not appear. So there is no doubt whatsoever that any effect of the universe can be explained satisfactorily because of its final causes, through the help of the method of maxima and minima, as can be by the very causes taking place… ‘
Leonhard Euler
I th t diti l R l i h Rit th d th i t l ti f ti h t b
(Basel,1707- Saint Petersburg,1783)
• In the traditional Rayleigh-Ritz methods the interpolating functions have to bedefined over the entire domain and have to satisfy the boundary conditions.
• Meanwhile in the FEM the interpolating trial functions are defined on a finite elementbasis, being more versatile when the shape is not simple enough
• The limitation is that the FEM trial functions have to satisfy in addition someconvergence conditions (continuity and completeness and compatibility)convergence conditions (continuity and completeness and compatibility)
finite elements in fluidsvariational approachvariational approach computational fluid dynamics I
• When using a variational approach, the aim is to find the vector function ofunknowns, that makes a minimum or a maximum of the functional I (typicallythe energy)
the energy)
dSx
gdVx
FI
,...,,...,
• After the discretezation has been carried out in terms of E smaller parts thepiecewise approximation is introduced so that
or in terms of the so called shape functions Ni
e
aproxe
where are the values of the unknowns at the nodes
···
ee
NNe2211
i
finite elements in fluidsvariational approachvariational approach computational fluid dynamics I
• Afterwards, the condition of extremezation of I with respect to i is imposed
II 1
0
M
i
I
II···
2
• Adding all those element contributions it is obtained
E e
• Assuming I to be a quadratic functional of the element equation results in
E
e i
e
i
II1
0
• Assuming I to be a quadratic functional of the element equation results in
eee
e
e
PKI
finite elements in fluidsvariational approachvariational approach computational fluid dynamics I
• After the assembling process it is obtained
PΦK
where and
E
e
e
1KK
E
e
e
1
PP
• After applying the boundary conditions the system is solved for the nodalunknowns iunknowns i
• Once i are known, we can obtain other variables as a post-processing value
finite elements in fluidsvariational approach, examplevariational approach, examplecomputational fluid dynamics I
• Example. Find the velocity distribution of an inviscid fluid flowing trough avarying cross section pipe shown in the figure
The governing equations are defined by finding the potential that minimizes the– The governing equations are defined by finding the potential that minimizes theenergy integral equation
dxddAI
L
0
2
21
with the boundary condition u(x=0)=u0, where the cross section area is
dx 02
LxeAA 0
A
u0
A0A1 A2
1 2 3
L
1 2 3
l(1) l(2)
L
finite elements in fluidsvariational approach, examplevariational approach, examplecomputational fluid dynamics I
• 1st step. Discretization
Divide the continuum into two finite elements. The values of the potentialfunction at the three nodes will be the unknowns of the femfunction at the three nodes will be the unknowns of the fem
• 2nd step. Select an interpolation model, easy but leading to convergence
The potential function will be taken as linearThe potential function will be taken as linear
bxax
and can be evaluated at each element as
x
where l(e) is the length of the e element
eeeee
lxx 121
g
finite elements in fluidsvariational approach, examplevariational approach, examplecomputational fluid dynamics I
• 3rd step. Derivation of stiffness matrices K(e) and load vectors P(e) by usinga variational principle
Deriving the interpolating function with respect to it is obtainedDeriving the interpolating function with respect to x it is obtained
eee
lo
eeeel eele xAdxAdxdAI 2112
122
12
2
2
122
oee
lldx 222 200
T
eeeee AA 11112 1122
12
2
where the cross sectional areas can be taken for the first and second element
eeTe
eeee
ee
lA
lAI ΦKΦ
21
1111
21
22
2
121
1212
e e t e c oss sect o a a eas ca be ta e o t e st a d seco d e e e tas and
where the nodal unknowns are respectively and2
10 AA 2
21 AA
11)(Φ
22)(Φ
2
3
finite elements in fluidsvariational approach, examplepp pcomputational fluid dynamics I
• 3rd step. (cont)
The minimal potential energy principle gives , if we take into account theexternal inflow
0
i
I
external inflow
where Q is the mass flow rate across section
eTeeeTeeeee
e
eeeee QQ
lAI QΦΦKΦ
21
22
221112
21
22
AuQ where Q is the mass flow rate across section
therefore, if we derive the functional I for each basic element
22 e AAI
AuQ
022 121221
112
21
22
111
)()()( eee
eeeeeeeee
e Ql
AQQlAI
02
22
)()()( eeeeeeeeeeee
QAQQAI
or in matrix form
022 212221
21212
221
)()()( eeee
eeeeeeeee Q
lQQ
l
1 eI 0
21
eeeeTeeeTe
ii
eI QΦKQΦΦKΦ
finite elements in fluidsvariational approach, examplevariational approach, examplecomputational fluid dynamics I
• 4th step. Assembly of the stiffness and load vectors
Once we have obtained the matrices for all the basic elements as
1111
1
11
lAK
1111
2
22
lAK
0111 uA
Q
23
2 0uA
Q
we can assemble the system to obtainQKΦ
0012211
1
1
1
1
0
0uA
AAAAlA
lA
223
2
2
2
2
2
2211 0
0uA
lA
lA
llll
finite elements in fluidsvariational approach, examplevariational approach, examplecomputational fluid dynamics I
• 5th step. Resolution of the system
As we need a reference value for the potentials (u3 is an unknown) we can set equal to 0 equal to 0
Taking A(1) as 0.80 A0 and A(2) as 0.49 A0, and l(1)= l(2)=L/2, the system of twoequations with two unknowns givesequations with two unknowns gives
• 6th step. Computation of the resultsLu01 651. Lu02 0271.
p p
Once we have obtained the potentials, the velocities can be derived by usingthe equivalence
12d
which gives the velocities at elements 1 and 2 as
112
ldxdu
0
1 251 uu . 0
2 052 uu .
finite elements in fluidsweighted residualsweighted residuals computational fluid dynamics I
• In this method the FE equations can be directly obtained from the governingequations (or equilibrium equations)
GF
The discretization is made and the field variable is approximated as
GF
n
xNx~
where i are constants and Ni(x) are linearly independent functions chosensuch that the boundary conditions are satisfied
i
ii xNx1
y
• A quantity R known as the residual or error is defined as
~~ FGR • The weighted function of the residual is taken as
FGR
0 dVRwfwhere f(R)=0 when R=0
0 dVRwfV
finite elements in fluidsweighted residualsweighted residuals computational fluid dynamics I
• There are several approaches to the weighted residuals method such as thecollocation method, the Least Squares method and the most commonly used ofall the Galeking methodall, the Galeking method
• In the Galerkin method the weighting functions are chosen to be equal to thetrial functions and f(R) is taken as Rf( )
with i=1,2,…,n0 dVRN
Vi
• In the rest of the aspects the method is similar to the variational
finite elements in fluidsweighted residualsweighted residuals computational fluid dynamics I
• Example. Find the velocity distribution of an inviscid fluid flowing trough avarying cross section tube shown in the figure
The governing equations are given by the continuity equation– The governing equations are given by the continuity equation
02
2
dd
with the boundary condition u(x=0)=u0, where the cross section area is
2dx
LxeAA 1
u0
A1A2 A3
u0 1 2 3
l(1) l(2)
L
finite elements in fluidsweighted residualsweighted residuals computational fluid dynamics I
• 1st step. Discretization
Divide the continuum into two finite elements. The values potential function inthe three nodes will be the unknowns of the femthe three nodes will be the unknowns of the fem
• 2nd step. Select an interpolation model, easy but leading to convergence
The potential function will be taken as linearThe potential function will be taken as linear
bxax
and can be evaluated at each element as
x
where l(e) is the length of element e
eeee
lxx 121
g
finite elements in fluidsweighted residualsweighted residuals computational fluid dynamics I
• This can also be obtained through the shape functions which have to be 1 at itsnode and zero at the others, that is
xN 11
1
xNxNx 2211
elN 11
x
l(e)
this is
elxN 21
l(e)
this is
eee lx
lx
lxx 12121 1
(the same as obtained before)
finite elements in fluidsweighted residualsweighted residuals computational fluid dynamics I
• 3rd step. Derivation of stiffness matrices K(e) and load vectors P(e) by usingequilibrium. Obtaining of a weak form
The integral of the weighted residual isThe integral of the weighted residual is
00 2
2
dxdxdw
el
i
integrating by parts dxdxddv 2
2dxdv
iwu idwdu
0002
dxdwddwldlwdwddwdxdw ile
ell
l eee
e 0000
00 2
dx
dxdxdxw
dxlwdw
dxdxwdx
dxw iiiii
12 0 uwulwdxddw
ie
i
l ie
120 dxdx ii
finite elements in fluidsweighted residualsweighted residuals computational fluid dynamics I
• This is
Th l t t i lt i t
122
121
02211
00 uwulwdx
dxdN
dxdN
dxdwdxNN
dxd
dxdw
ie
i
l il iee
• The elementary matrices result into
• As
h
0 eee PΦK
where
dxdNdNdNdNdx
dNdx
dNdx
dNdx
dN
dxdx
dNdx
dNdNdx
dNe
2212
2111
21
2
1
K
2
1
uueP
• As the derivatives are
the elementary matrices result into
dxdxdxdxdx
2
LdxdN 11
LdxdN 12
the elementary matrices result into
111
11
1122 leee dxll
e
K
11)(Φ
22)(Φ
1111 0
22
e
ee
ldx
ll
K
2
Φ
3
Φ
finite elements in fluidsweighted residualsweighted residuals computational fluid dynamics I
• 4th step. Assembly of the stiffness and load vectors
Once we have obtained the matrices for all the basic elements as
11111
11
lK
11111
22
lK
001 u
P
2
2 0u
P
we can assemble the system to obtainQKΦ
0111
01111
011ull
23
2
22
2211 0
110u
ll
llll
finite elements in fluidsweighted residualsweighted residuals computational fluid dynamics I
•5th step. Resolution of the system
6th t C t ti f th lt•6th step. Computation of the results
A b th t f ti bt i d b th i ht d id lAs can be seen, the system of equations obtained by the weighted residualsmethod is the same as in the variational method except for the absence of thedensity (which can be removed as it is a constant), and the cross section areas.y ( ),
The areas are not present in the second formulation as the system is solved invelocities and not in flow rates. To avoid this fact a two dimensional modelshould be considered.
finite elements in fluidsdiscretizationdiscretization computational fluid dynamics I
• Finite elements = Piecewise approximation of the solution by dividing the region into small pieces
• This approximation is usually made in terms of a power series (polynomial) which is easy to integrate and easy to be improved in accuracy by increasing the order, fitting in this way the shape of the polynomial to that of the solution (see figure)fitting in this way the shape of the polynomial to that of the solution (see figure)
• When the polynomial is of higher order (bigger than one) the midside and/or interior nodes have to be used in addition to the corner nodes
• Some other approximations such as Fourier series could also be used
• Problems involving curved boundaries can be solved using ‘isoparametric’ elements which are not straight-sided
finite elements in fluidsdiscretizationdiscretization computational fluid dynamics I
finite elements in fluidsdiscretization
• The mesh can be improved by
discretization computational fluid dynamics I
• The mesh can be improved by– Subdividing selected elements (h-refinement)– Increasing the order of the polynomial of selected elements (p-refinement)– Moving node points (r-refinement)– Defining a new mesh
• In higher order elements the midside and/or interior nodes have to be used in gaddition to the corner nodes in order to match the number of nodal degrees of freedom with the number of constants
• As it will be shown a different interpolation for the velocity and pressure• As it will be shown a different interpolation for the velocity and pressure unknowns is required for fem in fluids
• Basic elements to be considered– Triangular linear– Quadrilateral linear– Triangular linear (natural)g ( )– Triangular quadratic
finite elements in fluidsdiscretization, convergence
• The FEM is an approximation that converges to the exact solution as the element
discretization, convergence computational fluid dynamics I
• The FEM is an approximation that converges to the exact solution as the element size is reduced if:
i Th fi ld i bl d it d i ti t h t ti th l ti. The field variable and its derivatives must have representation as the element size reduces to zeroFor example, second derivatives cannot be represented with linear functionsThen the elements are said to be ‘complete’Then the elements are said to be complete
ii. The field variable and its derivatives should be continuous within the element (Cr
piecewise differentiable where r is the maximum order of derivatives within thepiecewise differentiable, where r is the maximum order of derivatives within the integrand)
dxdxd
r
r
(The polynomials are inherently continuous and satisfy this requirement)The field variable and its derivatives, up to the r-1-th, must be continuous at the element boundarieselement boundariesThen the elements are said to be ‘compatible’ or ‘conforming’
finite elements in fluidsdiscretization, convergencediscretization, convergence computational fluid dynamics I
• If we had for instance, ‘flat penthouses’ as interpolating functions, theinterpolating surface would be discontinuous (would ‘break and split up’)
• Still, there are many fem basic elements that not verifying the former properties still provide meaningful solutions (such as the ‘checker board pressure mode’)
100.00
25.00
30.00
35.00
40.00
45.00
50.00
55.00
40.00
50.00
60.00
70.00
80.00
90.00
5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00
5.00
10.00
15.00
20.00
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.000.00
10.00
20.00
30.00
finite elements in fluidsdiscretization, triangular linear b.e.
L t th b i li t i l l t ti d 1 2 d 3 b
discretization, triangular linear b.e.computational fluid dynamics I
• Let the basic linear triangular element connecting nodes 1, 2,and 3 be
• The equation that gives the• The equation that gives the surface (plane) is
(1) 2
(1)
that leads to the following
yxyx 321 ,
3 gequations
131211 yx 1
3
yx
y
232212 yx
333213 yx
22 , yx
11, yx 33 , yxx
finite elements in fluidsdiscretization, triangular linear b.e.discretization, triangular linear b.e.computational fluid dynamics I
• The solution of the former system gives
3322111 21 aaaA
1
(2)
3322112 21 bbbA
1 33
22
11
111
21
yxyxyx
A
(2)where
3322113 21 cccA
132 yyb 321 yyb
231 xxc
312 xxc
23321 yxyxa
31132 yxyxa
123 xxc 213 yyb 12213 yxyxa
substituting (2) in (1) and rearranging terms it is obtained
finite elements in fluidsdiscretization, triangular linear b.e.discretization, triangular linear b.e.computational fluid dynamics I
• The interpolating function results
332211 ,,,, yxNyxNyxNyx
where
233223321111 21
21 xxyyyxyxyx
Aycxba
AyxN , 233223321111 22
yyyyyA
yA
y,
311331132222 21
21 xxyyyxyxyx
Aycxba
AyxN ,
(3)
122112213333 21
21 xxyyyxyxyx
Aycxba
AyxN ,
(3)• The shape functions take the value of 1 at its node and cero at the rest• These expressions are complicated and depend on x and y
finite elements in fluidsdiscretization, triangular linear
• For an A element matrix equal to
discretization, triangular linearcomputational fluid dynamics I
• For an A element matrix equal to
dxdyy
Ny
Nx
Nx
NA jijiij
e
A
• The integrals are
yy
NNNNNNNNNNNN
313121211111
dxdy
NLNNNNNNNNNNy
Ny
Nx
Nx
Ny
Ny
Nx
Nx
Ny
Ny
Nx
Nx
Nyyxxyyxxyyxx
e
e
333323231313
323222221212A
• As the integrand is a constant there is no need to integrate numerically
yyxxyyxxyyxx
333323231313
s t e teg a d s a co sta t t e e s o eed to teg ate u e ca y
dxdyxxxxyyyyxxyyxxxxyyyyxxxxyyyyxxyy
A e
e12312113
231
213
12232132312313322
232
32
24A
exxyysimA e2
122
214
finite elements in fluidsdiscretization, triangular linear
Th b i l t t i lt
discretization, triangular linearcomputational fluid dynamics I
• The basic element matrix results
22
2
122
21
123121132
312
13
12232132312313322
232
32
4xxyysim
xxxxyyyyxxyyxxxxyyyyxxxxyyyyxxyy
A ee A
• That now can be assembledin the stiffness matrix to yield
2
1
2
1
··· ff
in the stiffness matrix to yield
6
6
5
4
3
2
6
5
4
3
2
··· ffff
2
9
8
7
6
9
8
7
6
···fffff
9 1010 f
finite elements in fluidsdiscretization, triangular linear b.e.
Th d f i t ti th h f ti d th i d i ti
discretization, triangular linear b.e.computational fluid dynamics I
• The need of integrating the shape functions and their derivatives over the domain leads to the use of the natural (local) coordinates, which allows for an element based integration that simplifies the calculations
• The natural triangular system of referenced is defined with the linear dependent coordinates L1, L2, and L3 3
AAL 1
1 AAL 2
2 AAL 3
3
2P
A2 A1
A
where A is the area defined by the point P and the opposite side1
2A31321 LLL
where Ai is the area defined by the point P and the opposite side
• The shape functions for this triangular linear element are
ii LN 321 ,,i
finite elements in fluidsdiscretization, triangular linear b.e.
3
discretization, triangular linear b.e.computational fluid dynamics I
• The shape functions are in fact as seen in (3) 3
PA2 A1
Ayx jj 2
11
1
2P
A3AA
yxyx
ALN i
kk
jj
ii 22
11
21
jkkjjkkjii xxyyyxyxyxA
yxLyxN 21,,
• Or in matrix form
xxyyyxyxL 1
1 233223321 1111
yx
yx
xxyyyxyxxxyyyxyx
ALL
21
12211221
31133113
3
2
33
22
11
21
yxyxA
finite elements in fluidsdiscretization, triangular linear b.e.discretization, triangular linear b.e.computational fluid dynamics I
• The derivatives of L1, L2 and L3 being3
A AyyL 321
yyL 132
yyL 213
2P
A2 A1
A3
Ax 2 Ax 2
Axx
yL
2312
Axx
yL
2231
Ax 2
Axx
yL
2123
1Ay 2y Ay 2
finite elements in fluidsdiscretization, triangular quadratic
• For natural coordinates in triangles the same procedure can be used
discretization, triangular quadraticcomputational fluid dynamics I
• For natural coordinates in triangles the same procedure can be used except for the fact that one of the three coordinates is linear dependant and can be dropped from the integration leading to a change in the integration limitsg g
1321 LLL
12
1
0
1
0 2112
1
0
1
0
11 1 dLdLLLgdLdLyxfJ
dxdyyxfLL
,,,
where the jacobian determinant is
LLLLJ 111221
and the integral is
A
xxyyxxyyAyxyx
J24 231331322
1221
1 1 12 dLdLLLAdfL
120 0 2112 dLdLLLgAdyxf
,,
finite elements in fluidsdiscretization, triangular quadratic
• For an A element matrix equal to
discretization, triangular quadraticcomputational fluid dynamics I
• For an A element matrix equal to
122 dLdLyL
yL
xL
xLAdxdy
yN
yN
xN
xNA jijijiji
ije
A
• The integrals are yyyy
313121211111 LLLLLLLLLLLL
12
1
0
1
0
333323231313
3232222212121
2 dLdL
LLLLLLLLLLLLyL
yL
xL
xL
yL
yL
xL
xL
yL
yL
xL
xL
yyxxyyxxyyxx
AL
ee
A
• As the integrand is a constant there is no need to integrate numerically
333323231313
yyxxyyxxyyxx
s t e teg a d s a co sta t t e e s o eed to teg ate u e ca y
1 1
12123121132
312
13
12232132312313322
232
32
2
1
42 L
e dLdLxxxxyyyyxxyyxxxxyyyyxxxxyyyyxxyy
AA A
0 02
122
21
4xxyysim
A
finite elements in fluidsdiscretization, triangular quadratic
• Integrating the basic element area it is obtained
discretization, triangular quadraticcomputational fluid dynamics I
• Integrating the basic element area it is obtained
21
21
121
11
1
0 1
1
0 1102
1 1
121
1
LLdLLdLLdLdL LL
(it is a half of the area of the square)
22 0
110 10 1020 0
12
(it is a half of the area of the square)• The integrals are
22
2
122
21
123121132
312
13
12232132312313322332
4xxyysim
xxxxyyyyxxyyxxxxyyyyxxxxyyyyxxyy
A ee A
which, as can be seen gives the same result as the one obtained with the global coordinatesthe global coordinates
finite elements in fluidsdiscretization, quadrilateral linear b.e.
• For linear quadrilateral elements The unknown function is now
discretization, quadrilateral linear b.e.computational fluid dynamics I
• For linear quadrilateral elements The unknown function is now expressed as
jh N ,,
4
where the quadrilateral natural coordinates are given by the lines that join the midpoints of opposite lines
jj
,,1
join the midpoints of opposite lines• The shape functions for this triangular quadratic elements result
1141
1 N
1
1141
3 N
1
1
2 (1,1)(-1,1)
1141
4
N 1141
2
N 3
4(-1,-1)(1,-1)
finite elements in fluidsdiscretization, quadrilateral linear b.e.
• The derivatives that show up in the integrals leading to the basic
discretization, quadrilateral linear b.e.computational fluid dynamics I
• The derivatives that show up in the integrals leading to the basic matrices are made in terms of the global coordinates
where the jacobian determinant is
ddgddJyxfdxdyyxf ,,,
where the jacobian determinant is
yxyxJ
J
finite elements in fluidsdiscretization, quadrilateral linear b.e.
• f(x y) is a function of N and its derivatives where its derivatives with
discretization, quadrilateral linear b.e.computational fluid dynamics I
• f(x,y) is a function of Ni and its derivatives, where its derivatives with respect to global coordinates can be written in terms of the local coordinates as
yNyNN iii 1
xNxNN
yyJx
iii
iii
1
where Cartesian and natural coordinates are related as follows
Jy
4
1ikk xNx
k
kNxx
k
kNxx
4
1kkk yNy
k
kNyy
k
kNyy
finite elements in fluidsdiscretization, quadrilateral linear b.e.
• For a A matrix equal to
discretization, quadrilateral linear b.e.computational fluid dynamics I
• For a A matrix equal to
ddJ
yN
yN
xN
xNdxdy
yN
yN
xN
xNA jijijiji
ijA
eNeNeNeNeNeNeNeNeNeNeNeNeNeNeNeNy
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
4141313121211111
dd
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eNy
N
y
N
x
N
x
N
y
N
y
N
x
N
x
N
y
N
y
N
x
N
x
N
y
N
y
N
x
N
x
N
J
4343333323231313
4242323222221212
A
• The integrals are
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eNyyyyyyyy
4444343424241414
e teg a s a e
4
1
4
1
4
1
4
1
1k K
kk
jkk
j
k K
kk
ikk
iij
NyNNy
NNyNNyNJ
A
ddNx
NNxNNxNNxN
k K
kk
jkk
j
k K
kk
ikk
i
4
1
4
1
4
1
4
1
finite elements in fluidsdiscretization, quadrilateral linear b.e.
• The integration of the elementary matrices has to be done in terms of
discretization, quadrilateral linear b.e.computational fluid dynamics I
• The integration of the elementary matrices has to be done in terms of a numerical procedure the most common of which is the Gauss integration. The integrals are evaluated as
n n1 1
i jijji fHHddfI
1 1
1
1
1
1 ,,
Hi=Hjn ij
1
1.00.577352
2.001 -1 1
for instance, if a four point Gauss rule is used it is obtained
0.888880.55555
00.77459
3
-1
• A Gauss surface integration with nxn Gauss points will be enough to bt i th t l ti f l i l f d t 2 1 i h
570570570570570570570570001001 .,..,..,..,.).)(.( ffffI
obtain the exact solutions for polynomials of grade up to 2n-1 in each direction of the space.
finite elements in fluidsdiscretization, triangular quadratic
• When the basic triangular element is quadratic
discretization, triangular quadraticcomputational fluid dynamics I
• When the basic triangular element is quadratic
2
45
1
3
y6
22 , yx
11, yx 33, yxx
• It is more convenient to express the shape functions in terms of the local (or natural) coordinates which are referred to every single basic elementnatural) coordinates which are referred to every single basic element.
finite elements in fluidsdiscretization, triangular quadratic
Th k f ti i d
discretization, triangular quadraticcomputational fluid dynamics I
• The unknown function is now expressed as
662211 NNNyx ···,
• The shape functions for this triangular quadratic elements result
662211y,
12 iii LLN214 4 LLN
325 4 LLN 1
• The jacobian determinant being
321 ,,i316 4 LLN
• The jacobian determinant being
NyNxNyNxyxyxJ k
kk
kk
kk
k
12211221 L
yLL
yLLLLL kkkk
finite elements in fluidsdiscretization, triangular quadratic
• For a A matrix equal to
discretization, triangular quadraticcomputational fluid dynamics I
• For a A matrix equal to
12dLdLJy
NyN
xN
xNdxdy
yN
yN
xN
xNA jijijiji
ij
A
6161···21211111y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
12
············
6262···22221212 dLdL
eNeNeNeNeNeNeNeNeNeNeNeN
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eN
y
eN
y
eN
x
eN
x
eNyyyyyy
J
A
• The integrals are
6666···2&261616y
N
y
N
x
N
x
N
y
N
y
N
x
N
x
N
y
N
y
N
x
N
x
N
e teg a s a e
4
1
4
1 221
4
1
4
1 1221
1k K
kk
jkk
j
k K
kk
ikk
iij L
NyN
LNy
LN
LNy
LN
LNy
LN
JA
12
4
1
4
1 1221
4
1
4
1 1221
dLdLLNx
LN
LNx
LN
LNx
LN
LNx
LN
k K
kk
jkk
j
k K
kk
ikk
i
finite elements in fluidsdiscretization, triangular quadratic
• For elementary triangles the numerical integration in terms of the
discretization, triangular quadraticcomputational fluid dynamics I
• For elementary triangles the numerical integration in terms of the natural coordinates gives
n
iii LLLfwdLLLfI 321321 ,,,,
ii LLLfwdLLLfI
1321321 ,,,,
Order Triang. Coord. Weights
Linear (O(h2)) 31
31
31 ,, 1
Quad, (O(h3))
021
21 ,,
31 Quad, (O(h )) 22
21
210 ,,
210
21 ,,
3
31
31
for instance, if a four point Gauss rule is used it is obtained
22 3
11111111
210
21
21
2100
21
21
31
21
321 ,,,,,, ffLLLfI
finite elements in fluids2D potential flow2D potential flowcomputational fluid dynamics 1
• The governing equation of the two dimensional potential flow is therefore given by
22
in 02
2
2
22
yx
where the velocity components are given by
with the boundary conditions
xu
yv
with the boundary conditionsDirichlet in
Newman in 2
0
0VnnV yxn
2
were nx and ny are the direction cosines of the outward unit vector n to 2
0yxn yxn
finite elements in fluids2D potential flow, Galerkin approach2D potential flow, Galerkin approachcomputational fluid dynamics 1
1.- Divide the region into E finite elements of p nodes each2.- Assume a suitable interpolation model for e in element e as
p
3.- Set the integral of the weighted (with weights equal to the interpolation
p
i
eii
e yxNyx1
,,
functions (Galerkin)) residue over the regions of the elements equal to zero
i=1,2,…,p
02
2
2
2
dyx
NIee
i
, , ,p
4.- Integration by parts (Green-Gauss theorem)
yx
0
dny
nx
Ndyy
Nxx
Nee
y
e
x
e
i
ei
ei
finite elements in fluids2D potential flow, green-gauss integ.
f
2D potential flow, green gauss integ. computational fluid dynamics I
The first second derivative in the integral, i.e.
n ny
dx
NI i 2
2
can be integrated by parts making dyd
xL
xR dyudvdxdyNI i
yB
yT yy
xxi
iNu x
dxxx
v
this results in with
xyB
dxdyxx
Ndyx
NI iy
y
x
xi
T
B
R
L
dndy x
and therefore
02
dnNdNdNee
ie
d
xxNdn
xNI i
xi
or Carrying out the same procedure for the derivatives with respect to y…
02
eee
dnx
Ndxx
dx
N xii
i
finite elements in fluids2D potential flow, integral equation
and writing the contour integral in terms of the boundary conditions as
2D potential flow, integral equation computational fluid dynamics I
and writing the contour integral in terms of the boundary conditions as
20
dVNdny
nx
Neee
iy
e
x
e
i
the integral equation results
221
therefore the Newman boundary conditions are naturally introduced
20
2
dVNdyy
Nxx
Nee
i
ei
ei
therefore, the Newman boundary conditions are naturally introduced
In matrix form eee PK
with
dTe BBK 20
2
dNVPTe
finite elements in fluids2D potential flow, matrix formulation2D potential flow, matrix formulation computational fluid dynamics I
where
31312121
2
12
1 NNNNNNNNNN
NNNx
Nx
Nx
N
321
321
B
3232
2
22
2
yN
yN
xN
xN
yN
xN
yyxxyyxxyx
T BB
yyy
2
32
3
yN
xNsim
yyy
.
(for a linear triangular basic element)Assembling the elementary matrices it is obtained
PK
finite elements in fluids2D potential flow, matrix formulation2D potential flow, matrix formulation computational fluid dynamics I
for a linear triangular basic element the elementary stiffness matrix results in
22
1 kikijijiii ccbbccbbcb1
22
22
41
kk
kjkjjj
kikijijiii
eTe
cbsimccbbcb
AdBBK ycxba
AN iiii
21
and the source matrix
011
20
20
2
ijTe sVdNVP
when the fluid is entering the edge ij of length sij with a velocity V0 normal to the edge
02
If the fluid is entering the edges jk or ik the elementary vectors are
10
0 jke sVP
01
0 ike sVP
11
2P
10
2P
finite elements in fluids2D potential flow, example
Let us apply the so-obtained formulation to the obtaining of the confined flow
2D potential flow, example computational fluid dynamics I
et us app y t e so obta ed o u at o to t e obta g o t e co ed oaround a cylinder. Due to its symmetry, the domain can be chosen to be a forth of the total area
2 3 41
0 n
3
42
16
13
5
2 3
6
7 8
1
0
7
8 9
12
10
1113
5 6
12131 n
0
Th b d diti ill b
8 9
9 10 11
0 nThe boundary conditions will be:•An inlet constant velocity V0=1 normal to boundary 1-9•Constant potential in boundary 4-13 due to the symmetry (taken as cero for convenience
0 n
convenience •Tangential velocity (V0=0) in the rest of the boundaries
finite elements in fluids2D potential flow, example
For the nodes coordinates shown the node-1 basic stiffness matrices are
2D potential flow, example computational fluid dynamics I
o t e odes coo d ates s o t e ode bas c st ess at ces a eobtained as
Nodo x yNodo x y
1 0 8
2 5 8
3 9.17 8
0406250400251
1 ....
K4 12 8
5 0 4
6 5 4
7 9 17 5 5
6250.sim
7 9.17 5.5
8 12 5.5
9 0 0
10 5 0
022
1P11 8 0
12 9.17 2.83
13 12 4
0
finite elements in fluids2D potential flow, example
N
2D potential flow, example computational fluid dynamics I
Notes
•As the potentials 4= =0, the corresponding rows and columns can be li i t d f th t f ti t b l d f eliminated from the system of equations to be solved for 1, 5, ,
9, , 12.•The velocity can be obtained as a post-process value making
ij
ijji xxx
u
ij
ijji yyy
v
finite elements in fluids2D laminar NS, preliminary issues
• Up to this point some simple flow problems have been solved but there are
2D laminar NS, preliminary issuescomputational fluid dynamics I
• Up to this point some simple flow problems have been solved, but there are many flow problems in which the navier-stokes equation has to be solved
• The navier-stokes equations present two important problems that did not arise before i.e.:
– The presence an heterogeneous set of unknowns (velocity and pressure) that requires special treatment.This has to do with the verification of the continuity equation and causes instabilities in the pressure field, no matter the Reynolds number is.Therefore a proper combination of the velocity and pressure field is required p p y p qunless special formulation to circumvent this is added
– The non-linear convective term appears, turns the stiffness matrix into a non-symmetric one and leads to a certain amount of instability. y yThe Galerkin formulation lacks stability when convective effects dominate (this is for large Reynolds numbers, Re=uL/v) and alternative stabilizing techniques have to be used (Petrov-Galerkin, Characteristics Galerkin, GLS, SUPG, Finite Increment Calculus, Bubble functions,...)
finite elements in fluids2D laminar NS, velocity & pressure
• The need of verifying the continuity condition (divergence free
2D laminar NS, velocity & pressurecomputational fluid dynamics I
• The need of verifying the continuity condition (divergence free condition) together with the existence of two different types of variables is mainly solved in two different ways:
– Keeping both variables and both sets of equations leads to the so-called mixed formulation, the stiffness matrix becomes a partitioned one with a null submatrix in the diagonal and a proper selection of theone with a null submatrix in the diagonal and a proper selection of the basic elements has to be made to verify the LBB condition or circumvent itTh lt f l ti t k– The penalty formulation takes
instead of the continuity condition, where is a very small parameter, p u·
y , y p ,and takes it into the dynamic equation.The selection of is a difficult task as being too small may cause a loss of accuracy and being too big may prevent the convergenceof accuracy and being too big may prevent the convergence
finite elements in fluids2D laminar NS, algebra issues2D laminar NS, algebra issues computational fluid dynamics I
• Some kind of functions to which the velocity and pressure fields must belong
S b l f i t bl f ti ithi 2L
Some kind of functions to which the velocity and pressure fields must belongto are presented
• Sobolev space of square integrable functions within
• Hilbert space, subspace of L2 of functions whose
2L
kHderivatives up to the k-th also belong to L2
• Subspace of L2 with a null mean in the domain (to beused in relation to the pressure. Can be avoided by
20L
p ysetting a given pressure at a certain point)
• Subspace of functions that, belonging to H1, arecancelled at the boundary
10H
y
finite elements in fluids2D laminar NS, algebra issues
• With respect to the velocity u the space of trial solutions is denoted by
2D laminar NS, algebra issues computational fluid dynamics I
With respect to the velocity, u, the space of trial solutions is denoted byS, which must be a subspace of H1 that satisfies the Dirichletconditions on the boundary
• The weighting functions of the velocity, w, will belong to the V spaceswhich belong to H1 and vanish on the boundary where the velocity isprescribed
• Finally the subspace Q will be introduced for the pressure. As noli it b d diti ib d f dexplicit boundary conditions are prescribed for pressure and no
derivatives of the pressure show up in the weighted formulation as itwill be seen it is only required that Q belongs to L2 for both the trialwill be seen, it is only required that Q belongs to L for both the trialand weighting functions
finite elements in fluids2D laminar NS, weighted residuals2D laminar NS, weighted residuals computational fluid dynamics I
• Once we´ve got the governing Navier-Stokes differential equation, we aregoing to solve it by using the finite element method, that is, to obtain theapproach
n
Nˆ
where Ni are the shape functions defined on a local basis for each element
ii1
Nuu
where Ni are the shape functions defined on a local basis for each elementand where the coefficients i are the unknowns for each of the nodes
• In order to do so we need to express the governing equation as• In order to do so, we need to express the governing equation as
m
ejjjj eedddd
1gGuguG
so as to obtain the integral forms, we can apply the weighted residualsmethod (de Galerkin) or the variational( )
finite elements in fluids2D laminar NS, weighted residuals
The 2D equations to be solved are
2D laminar NS, weighted residuals computational fluid dynamics I
• The 2D equations to be solved are
01 ijjiijijti fupuuu ,,,,
• Multiplying the equation by the weighting functions and integrating within the
0iiu ,
Multiplying the equation by the weighting functions and integrating within thedomain, it is obtained
01
dfupuuuw ijjiijijtii ,,,,
0dqu ii ,
if the these integral expressions are verified for any wi, q, then, the differentialequations will be satisfied within the whole domain
finite elements in fluids2D laminar NS, weighted residuals
• The viscous term is
2D laminar NS, weighted residuals computational fluid dynamics I
• The viscous term is
dyu
xuwduw ii
ijjii 2
2
2
2
,
• Integrating the first term by parts(in 2D)
dyudvdxdyuwI ii
y yyxxi
iwu udxuv ii
n n
dyd
xL
• The integral b.p. results into
i xxx
yT
xLxR
x
xyBdxdy
xu
xwdy
xuwI ii
y
y
x
x
ii
T
B
R
L
uwu
d
xu
xwdn
xuwI ii
xi
i dndy x
finite elements in fluids2D laminar NS, Green theorem
• Proceeding in an analogous way with the second term it is obtained
2D laminar NS, Green theorem computational fluid dynamics I
Proceeding in an analogous way with the second term, it is obtained
duwdnuwduw iiy
ii
ii
therefore
yyyyy yii
dnuwduwduw jjiijijijjii ,,,,
• Proceeding in an analogous way with the pressure term
dpnwdpwdpw iiiiii 111
,,
finite elements in fluids2D laminar NS, weak form
• The stationary formulation that also ignores the convective terms gives
2D laminar NS, weak formcomputational fluid dynamics I
• The stationary formulation that also ignores the convective terms gives
21
dwtdfwpdwduw iiiiiijiji ,,, 2
2
iiiiiijiji ,,,
0dqu ii ,
Vwi Qq ii bu 1 ijij tn 2
or in vectorial notation
21
dddpd wtfwwwu ···: 22
dddpd wtfwwwu :
0dq u·
q
finite elements in fluids2D laminar NS, matrices2D laminar NS, matricescomputational fluid dynamics I
• The Stokes simplification can be written as
y
x
TTy
x
ff
pvu
BBBABA
where
yx pBB
dNwNwA jiji
ij
dwB j
ixij
dwB ji
yij
e yyxxij e x jxij e y jyij
dtwdfwf xiixiixi
ee
dtwdfwf yiiyiiyi ee ee
finite elements in fluids2D laminar NS, LBB
• As stated before, the selection of the basic functions N and is not trivial, as it
2D laminar NS, LBBcomputational fluid dynamics I
As stated before, the selection of the basic functions N and is not trivial, as it involves different types of equations and unknowns
• The matrix expression of the stationary flow results into
hfu
0BBK
T
nxn nxm
hp0BT
mxn
where the dimension of K is nxn (n range) and the dimension of B is nxm. Inorder to obtain a unique solution, the stiffness matrix (with dimensions(n+m)x(n+m)) must be non-singular and in order to achieve so, it is required(although not sufficient) that
in other case the range of matrix B would be n (its range cannot be bigger than
hh VQ dim dim
n) there would be more equations than unknowns, and the system would benon-compatible
finite elements in fluids2D laminar NS, LBB
• The sufficient condition is the so called LBB (Ladyzhenskaya -Babuska-
2D laminar NS, LBB computational fluid dynamics I
The sufficient condition is the so called LBB (Ladyzhenskaya BabuskaBrezzi) condition, that sets that for a given value of regardless of themesh size, the existence of an approximate solution (uh, ph), requires anelection for subspaces Vh, Qh, so that
010
hi
hii
h
VwQq wq
dwqhh
ihh
,supinf
from the system fBKu p
0uBT
solving first equation for u gives
pBfKu 1
that once introduced at the second one gives
hfKBBKB 11 TT p hfKBBKB p
finite elements in fluids2D laminar NS, LBB
• If the velocity-pressure pair verifies the LBB condition that is it ensures
2D laminar NS, LBB computational fluid dynamics I
• If the velocity-pressure pair verifies the LBB condition, that is, it ensuresthat ker B = 0, the matrix BTK-1B is positive definite, the stiffness matrixis regular (non-null determinant) and the problem of finding
if i h i i i ll d i d
hh Vu hh Qp verifying the equations is univocally determined
• We are going to illustrate this condition with some particular examplesof velocity-pressure pairs. After that, the most popular pairs for fluidy p p , p p pproblems will be presented
• In order to do so the particular case of a square domain of n by n nodes(like the one being used to evaluate the cavity flow) will be presented(like the one being used to evaluate the cavity flow) will be presented
2 3 n..1 1350
400
Level vel19 0 95
2
3
:
Y
50
100
150
200
250
300
350 19 0.9518 0.917 0.8516 0.815 0.7514 0.713 0.6512 0.611 0.5510 0.59 0.458 0.47 0.356 0.35 0.254 0.23 0.152 0.11 0.05
nX0 50 100 150 200 250 300 350 400 450 500
0
finite elements in fluids2D laminar NS, LBB
• For a domain divided into P P (linear velocity/constant pressure)
2D laminar NS, LBB computational fluid dynamics I
• For a domain divided into P1P0 (linear velocity/constant pressure) triangular elements of nxn nodes, the number of equations/unknowns in the continuity equation is
Equations (pressure unknowns): 2(n-1)2-1Unknowns (velocities): 2(n-2)2
2 3 n..
2
1 1
3
n:
As the number of equations is bigger than the number of unknowns the only possible solution is the trivial and the problem is ‘locked’y p p
finite elements in fluids2D laminar NS, LBB
• For a domain divided into P P (quadratic velocity/linear pressure)
2D laminar NS, LBB computational fluid dynamics I
• For a domain divided into P2P1 (quadratic velocity/linear pressure) triangular elements of nxn corner nodes, the number of equation and unknowns in the continuity equation is
Equations (pressure unknowns): n2-1Unknowns (velocities): 2(2n-3)2
2 3 n..
2
1 1
3
n:
As the number of equations is smaller than the number of unknowns (n>2), the pair allows for a solution different from the trivial
finite elements in fluids2D laminar NS, LBB
• Similarly for a domain divided into Q P (bilinear velocity/constant
2D laminar NS, LBB computational fluid dynamics I
• Similarly, for a domain divided into Q1P0 (bilinear velocity/constant pressure) triangular elements of nxn corner nodes, the number of equations/unknowns in the continuity equation is
Equations (pressure unknowns): (n-1)2-1Unknowns (velocities): 2(n-2)2
11 2 3 n..
2
3
n:
As the number of equations is smaller than the number of unknowns (n>4), the pair allows for a solution different from the trivial, but not necessarily exact as it leads to a discontinuous checkerboard pressure modeexact as it leads to a discontinuous checkerboard pressure mode
finite elements in fluids2D laminar NS, LBB
S f th t l l it i
2D laminar NS, LBB computational fluid dynamics I
Some of the most popular velocity-pressure pairs are
• Q1P0 (bi-linear velocity – constant pressure);Q1P0 (bi linear velocity constant pressure)LBB is not verified , discontinuous pressure
• P1P0 (linear velocity - constant pressure)LBB is not verified , discontinuous pressure
• Q1Q1 (bi-linear velocity – bi-linear pressure)LBB is not verifieds ot e ed
• P1P1 (linear velocity - linear pressure)LBB is not verified
finite elements in fluids2D laminar NS, LBB2D laminar NS, LBB computational fluid dynamics I
• P2P1 (quadratic velocity – linear pressure)LBB is verified ;
• Q2Q1 ‘Taylor-Hood’(bi-quadratic velocity, bi-linear pressure)y ( q y, p )LBB is verified
• Q2Q1 ‘Taylor-Hood serendipity’ (bi-linear velocity and pressure)LBB is verified
finite elements in fluids2D laminar NS
• Taking the unsteady convective flow equations
2D laminar NScomputational fluid dynamics I
Taking the unsteady convective flow equations
012
2
dwtpdwduwdfuuuw iiiijijiijijtii ,,,,,
0dqu ii ,
Vwi Qq ii bu 1 jiji xuxu 00 ,
or in vector notation1
jj
1
u 012
2
dwtdpddt iiw·w:ufu·uuw
0udq ·
0udq
finite elements in fluids2D laminar NS, matrices2D laminar NS, matricescomputational fluid dynamics I
• When the transient convective flow is considered a finite different approach can be considered to integrate the differenctial equation. The derivatives with respect to time can be taken as
tt
nn 1 MM
where are the velocities obtained for the previous temporal step
11
1n
11,1
nnnnnnn
tt MfBpAvuCM
0B nT 0B T
finite elements in fluids2D laminar NS, matrices2D laminar NS, matricescomputational fluid dynamics I
• When the convective term is included a non-linear convective term is obtained that can be evaluated making use of Newton-Raphson or a successive approximation technique, the former leading to
nnnnnn 11 v,uCv,uC
where are the velocities obtained for the previous convection step and can be estimated are zero for the step
11 nn v,u
111 11
nnnnnnn
tt MfBpAv,uCM
• For each time step the iterations for convection should be carried out
0B nT
For each time step the iterations for convection should be carried out
finite elements in fluids2D laminar NS, matrices2D laminar NS, matricescomputational fluid dynamics I
• When the transient and convective flows are taken into account it is obtained
nnnn uv,uCuM 11
1
n
nnn
n
n
tpvv,uC
pvM 111
1
1
1
1
1
n
n
n
y
x
n
n
n
Ty
Tx
y
x
tpvu
MM
ff
pvu
BBBABA
where
NN jj
e
dNwM jiij
e
dy
NvN
xN
uNwC jkk
jkkiij
finite elements in fluids2D laminar NS, solver
• The direct solvers are one step methods that give an exact solution to the
2D laminar NS, solvercomputational fluid dynamics I
• The direct solvers are one-step methods that give an exact solution to the systems. Nevertheless, the memory requirements involved are very high even with skyline storing and specially for fluids
• The sparse storage allows to drop all the non zero elements but cannot be used• The sparse storage allows to drop all the non-zero elements but cannot be used in combination with a direct solver due to the fact that some elements could be ‘thrown out’ of the sparse stencil
• The row-indexed sparse storage mode requires a memory space of only twice the• The row-indexed sparse storage mode requires a memory space of only twice the number of the non-zero matrix elements uses an integer pointer vector (p) and a real vector (v), where the sparse elements are loaded
• An iterative solver of the Kryliov type such as the (PBCG or the GMRES) can be• An iterative solver of the Kryliov type, such as the (PBCG or the GMRES) can be used in connection with the sparse matrix storing
finite elements in fluids2D laminar NS, benchmark
• Benchmark problem: Cavity flow
2D laminar NS, benchmarkcomputational fluid dynamics I
10000Re /uL• Benchmark problem: Cavity flow 10000Re /uL
350
400
350
400
Level vel19 0.9518 0.917 0.85
Y 200
250
300
Y 200
250
300 16 0.815 0.7514 0.713 0.6512 0.611 0.5510 0.59 0.458 0.47 0 35
50
100
150
50
100
1507 0.356 0.35 0.254 0.23 0.152 0.11 0.05
X0 50 100 150 200 250 300 350 400 450 500
0
X0 50 100 150 200 250 300 350 400 450 500
0
finite elements in fluids2D laminar NS, benchmark
• Benchmark problem: Cavity flow 10000Re /uL
2D laminar NS, benchmarkcomputational fluid dynamics I
• Benchmark problem: Cavity flow 10000Re /uL
4 444
5 55 55 55
5 666 6
6 6
6
6
7
88
89 910
1111
12134 15 6 16
17400Re = 10000
1
22
22
2
3
33 3
33
3 3
33 3
3
4
4
4
4
4
4
4
44
4 4
5 5
55
5 5
5
5 5
6 66
6
66
7
77 7
8
Y 200
250
300
350Level h19 0.038030918 0.035016717 0.032002616 0.028988515 0.025974414 0.022960213 0.019946112 0.01693211 0.013917910 0.0109037
0.8
1
1.2
Ghia 129x129
1 1
1
2
2
2
2
3
3
3
3
4
444 4
444 4
4 4
5
5
5 5
5
5
5 5
5
5 5
5 5
666
6
6
6
6
Y
50
100
150
200 10 0.01090379 0.007889628 0.00487557 0.001861376 -0.0001703225 -0.001152754 -0.002867523 -0.004166882 -0.005503661 -0.0062906 0.2
0.4
0.6Ghia 129x129Present 41x41
56
66
6
66 66 6 6 67
X0 100 200 300 400 500
00-0.5 0 0.5 1 1.5
finite elements in fluids2D laminar NS, benchmark
• Benchmark problem: Backward step 1200Re /uD
2D laminar NS, benchmarkcomputational fluid dynamics I
• Benchmark problem: Backward step 1200Re /uD
6
7
8
X
Y
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Y 4
5
6
7
8
VEL0.9375680.8750640.8125590.7500550.687550.6250460.5625410.5000360 437532
X0 10 20 30 40 50
0
1
2
3
0.4375320.3750270.3125230.2500180.1875140.1250090.0625046
1 123 3 44
10 12 1
Y
2
3
4
5
6
7
8
15 -0.0061812714 -0.012678213 -0.019175212 -0.025672211 -0.032169210 -0.03866619 -0.04516318 -0.05166017 -0.05815716 -0.0646545 -0.0711514 -0.0776483 -0.0841452 -0.09064191 -0.0971389
11 1
11
11
2
22
2
2
3
3
33 3 3 3 3
444 4
44
44
555 5
5
6
6
7 88
8
8
99
9 9 99 9
9 9999
9
10
10 10
0
1111
1111
212
1212
313
13
14
14
151515
15
X0 5 10 15 20 25 30 35 40 45 50
0
1
finite elements in fluids2D laminar NS, benchmark
• Benchmark problem: Backward step 1200Re /uD
2D laminar NS, benchmarkcomputational fluid dynamics I
• Benchmark problem: Backward step 1200Re /uD
s3
s2
s1
Reattachment length s3
15
20
25
30
s3 Armaly exps3 Armaly cal
0
5
10
15 s3 Armaly calPresent
0 200 400 600 800 1000 1200 1400
finite elements in fluidspenalty formulation
Th lt th d b i t t d bli l ti f th
penalty formulationcomputational fluid dynamics I
• The penalty method can be interpreted as enabling a relaxation of the incompressibility constraint so that the incompressible problem is approached by a slightly compressible formulationapproached by a slightly compressible formulation
• The pressure unknown is removed from the mixed formulation and therefore the only unknowns of the problem will now the pressuresy
finite elements in fluidspenalty formulation
• The penalty formulation is based upon substituting the incompressibility condition
penalty formulationcomputational fluid dynamics I
• The penalty formulation is based upon substituting the incompressibility conditionby
pu ii ,
where tends to zero, and drive it into the dynamic equation to obtain
1 df
ti th t d t d d th k idi th
01
dwuwfwu iiiiiijiji )( ,,,,
equation that does not depend upon the pressure unknown, so avoiding theproblems found in the mixed formulation
finite elements in fluidspenalty formulation
• In order to impose the incompressibility condition is required to be very small
penalty formulationcomputational fluid dynamics I
• In order to impose the incompressibility condition, is required to be very small,but not to much, because in that case, the penalty term (if it leads to a regularmatrix) promotes the obtaining of a unique trivial solution
• In order to avoid it, it is required to carry out a ‘reduced integration’ or ‘selectiveintegration’, that is to carry out a roughly approximating of the penalized term,leading to a similar effect to that obtained for the mixed elements verifying theleading to a similar effect to that obtained for the mixed elements verifying theLBB condition
• It is recommended to chose as
with c=107 1 Rec ,max
• Q1P0 (2x2, reducida 1)
• P1P0 (1, reducida 1)
• P2P1 (3, reducida 1)
• Q2Q1 (3x3, reducida 2x2)
• Q2Q1 (3x3, reducida 2x2)
finite elements in fluidspenalty formulation, reduced integrat. computational fluid dynamics I
• Proceeding in the same way as in the mixed formulation it is obtained
or in matrix notation
012
2
dwtdwuduwfuuuw hi
hi
hii
hii
hji
hji
hi
hji
hj
hti
hi
h
,,,,,,
fBAvuCM
1,
t
y
x
yTx
t ff
vu
BDDB
vu
AA
vu
vuCvuC
vu
MM
1
,,
finite elements in fluidspenalty formulation, matrices computational fluid dynamics I
• Where the basic matrices are
n
n
n
n
nn
nn
n
n
t vu
AA
vu
vuCvuC
vu
MM
,,1
1
111n
n
y
xn
n
yTx
t vu
MM
ff
vu
BDDB
e
dNwM jiij
e
dy
NvN
xN
uNwC jkk
jkkiij
e
dy
Nyw
xN
xwA jiji
ij
e
dx
NxwB ji
yij
e
dy
NywB ji
xij
e
dy
NxwD ji
ij
ee
dtwdfwf hxiixiixi
ee
dtwdfwf hyiiyiiyi
finite elements in fluidspenalty formulation, matrices computational fluid dynamics I
• As we know, the Shallow Water equations are
finite elements in fluidsssww equationscomputational fluid dynamics I
34
2
2
2
2
2
h
xawc H
uVgnH
WWCyu
xuvf
xhg
yuv
xuu
tu
34
2
2
2
2
2
h
yawc H
vVgnH
WWCyv
xvuf
yhg
yvv
xvu
tv
0
yvH
xuH
th
• Following a similar procedure as the one carried out for the 2D laminar equations in the continuity equation it is obtained
finite elements in fluidsssww equationscomputational fluid dynamics I
0
yvH
xuH
th
dqHuduHqqh Niit ,,
0,,
dHuhq iit
dqHudHuqdqh iiit ,,
• Identically, with the dynamic equation, it is obtained
finite elements in fluidsssww equationscomputational fluid dynamics I
34
2
2
2
2
2
h
xawc H
uVgnH
WWCyu
xuvf
xhg
yuv
xuu
tu
iiijjijijt Sghghuuuu ,,,,,,
2,,,,,2
dwtdhwgduwdSuuuwh
hhh
hi
hi
hhii
hji
hjii
hji
hj
hti
hi
34
2
2
2
2
2
h
yawc H
vVgnH
WWCyv
xvuf
yhg
yvv
xvu
tv
• And the matrix expressions are
finite elements in fluidsssww equationscomputational fluid dynamics I
e
dy
NvN
xN
uNwC jkk
jkkiij
e
dy
Nyw
xN
xwA jiji
ij
e
dxwgB j
ixij
e
dywgB j
iyij
e
dNx
HD ji
kkxij
ee
dtwdSwf xiixiixi
ee
dtwdSwf yiiyiiyi
e
dNy
HD ji
kkyij
h
y
x
yx
y
x
hh fff
hvu
DDBABA
hvu
vuCvuC
,,
e
dHuqf Nihi
CFDI
finite elements in fluids
computational fluid dynamics I
stabilizing techniques
• The stabilizing techniques are ways of circunventing the LBB conditionthat allow to use velocity-pressure terms which are not stable for thestandard Galerkin formulation. These problems grow bigger as theconvective term is larger compared to the other terms in the equation
• The basic idea behind stabilization is to enforce the positive definitenessof the matrix
• The most commonly use of these methods are the Petrov-Galerkin,Characteristics Galerkin, GLS, SUPG, Finite Increment Calculus, BubbleFunctions,...
finite elements in fluidsstabilizing techniquescomputational fluid dynamics I
0BBK
T
• The stiffness matrix is non-symmetric and its treatment with the Galerkinfunctions is not adequate
• To show these problems we are first going to use the one dimensional steadyconvection-diffusion equation
• After dealing with the transport equation, the particulars will be generalized forthe multidimensional Navier-Stokes equations
• The former equation has an analytical solution for constant U and k, and Q equalto zero which is given by (if there is no convection it would be a straight line)
• For boundary conditions (x=0)=0 and (x=L)=1 the solution is given by
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
0
Q
dxdk
dxd
dxdU
xkU
ecc 21
kUL
xkU
e
e
1
1 k
ULk
UL
Lk
U
ee
eLx
1
1
1
122
/x
L
1
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• We can introduce the discretization
• For an internal element the following equation is obtained
where
• Integrating the elementary matrices elements, taking U, Q and k as constantsand the weighting functions equal to the shape functions it is obtained
0 ijij fK
h jih j
iij dxdx
dNk
dxdwdx
dxdN
Uwk00
h
ii Qdxwf0
211U
hkk
222U
hkk
212U
hkk
221U
hkk
21Qhf
22Qhf
hxN 11
hxN 2
kkN
1
i-1 i i+1
N1 N2
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• Assembling the matrices it is obtained
• And the equation for our internal element is
or
0
··121··
2
··
··
··········
··22
····
··2222
··
····22
············
1
1
QL
UhkU
hk
UhkU
hkU
hkU
hk
UhkU
hk
i
i
02
22 11
Qh
hkU
hkU
hk
iii
01212
11 kQhPePe iii
kUhPe2
• In fact, the so obtained solution is also the result of using a finite differentapproach. In finite differences the derivative is approached by the secant
• Taking the Taylor expansion of
a function f(x) around x as
• The first derivative can then
be approximated by
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
···´ xhfxfhxf
h
xfhxfxf
xx-h x x+h
f(x)y
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• Taking the third term in the Taylor´s expansions
adding the former expressions it is obtained
subtracting the former expressions it is obtained
• For the variables in our equation the central approximation results into
···´ xfhxhfxfhxf 2
···´ xfhxhfxfhxf 2
xfh
xfhxfhxf
222
hdxd ii
211
211
2
2
22hxd
d iii
xfh
hxfhxf
2
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• Substituting the central finite difference approaches into the governing equation,it is obtained
which is the same equation as the one obtained before for the Galerkinapproximation
• Pe, the Peclet number, measures the importance of the convection in theequation. The stiffness matrix is not symmetric and, the bigger the Peclet numberis, the more non-symmetric the matrix turns
0
Q
dxdk
dxd
dxdU
012122
2
2
1121111
kQhPePeQ
hk
hU iii
iiiii
kUhPe2
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• When the Peclet number tends to infinity (theconvection is dominant) the solution is purelyoscillatory and non-sense, this is shown in thepicture for the value = 0, which means aGalerkin formulation is being used.
(Q is taken as 0, and the boundary conditionsare x and xL
for Pe=0 (no convection) the differentialequation is d2dx2=0, the solution of which isa straight line, and for Pe infinite (no diffusion)the solution of the equation is ddx=0 is aconstant, and only a boundary condition canbe imposed (because the eq is of first order))
For strong convection the downstreamboundary condition is only noticed in avery small region and the upstreamweighting is more adequate !
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• As the propagation of the information is in the direction of the velocity, a straightforwards procedure would be to substitute the first central finite derivative by afull upwind differencing
• This results in an equation
• This full upwind solutions provide with realistic (though not always accurate)solutions for several Peclet numbers ( in the plot The exact solution beingonly obtained for infinity Peclet numbers (in the same way the Galerkinformulation obtains the exact solution only for full diffusion -> Petrov-Galerkin is amixture of both that achieves an exact solution for all Peclet numbers.
hdxd ii 1
02222
11 kQhPePe iii
211
2
2
22hxd
d iii
hdx
d ii 1 0U 0U
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• The Petrov-Galerkin methods arebased in taking the weightingfunctions equal to
where
*iii wNw
e
hdxwi 2*
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• If we take the simplest discontinuous function
the equation to be obtained for the so-posed problem is
when the full upwind formulation is obtained and when the Galerkinformulation is got
• It can be proved that the selection
gives the exact nodal values for all values of Pe.
• It can also be shown that with oscillations will take place whenever
UU
dxdNhsignU
dxdNhw ii
i 22 *
01122112
11 kQhPePePe iii
Peeeee
PePe PePe
PePe 11
coth
12
k
UhPe
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• The above weighting functions should be applied on any term in the equation
• The weighting functions can be taken as discontinuous as far as the convectiveterms are concerned
• But when the weak form of the diffusive term is taken into account the derivativeof the weighting function is required and the discontinuity can not take place atthe node but within the element (see figure)
dxdx
dNUw
h ji0
h ji dx
dxdN
kdxdw
0
finite elements in fluidsstabilizing techniques, conv-diffcomputational fluid dynamics I
• If instead of the governing equation considered we substitute it by the expression
with
that is, an artificial diffusion that acts in the direction of the flow is considered, itis obtained
which is the same expression obtained for the Petrov-Galerkin formulation
• This shows that the effects of considering a Petov-Galerkin formulation is at lastthat of considering an artificial diffusion that acts in the direction of the flow
0
Q
dxdkk
dxd
dxdU b
Uhkb 21
01122112
11 kQhPePePe iii
• The GLS (Galerkin least squares method) is a different approach that leads tothe same formulation
• In the GLS method the positive definiteness of the stiffness matrix isaccomplished by carrying out a modification in the weak form of theincompressibility condition rending non-zero diagonal terms in the stiffnessmatrix
• If we pose the same differential problem with the differential operator L notationthe previous considered problem can be written as
with
leading to the Galerkin approximation for the k-th equation
finite elements in fluidsstabilizing techniques, conv-diff computational fluid dynamics I
0QL Nˆ
dxdk
dxd
dxdUL
dxQLNL
k ̂0
• In the GLS method the square of the residual
has to be minimized, therefore
or
• If we take now a linear combination of the Galerkin formulation plus times theformer equation, it is obtained
00
dxQLN
dxdk
dxd
dxdNUN
L
kk
k ˆ
QLR ̂
021
00
2 dxdLdQLdxR
dd L
k
L
k
ˆˆ
021
00
2
dxN
dxdk
dxd
dxdNUQLdxR
dd L
kkL
k
ˆ
finite elements in fluidsstabilizing techniques, conv-diff computational fluid dynamics I
• If we drop the second-order term as we could do for linear shape functions andwe take
the formulation obtained is
i.e. the same as the one obtained for the Petrov-Galerkin formulation
020
dxQL
dxdN
UhU
NL k
k ˆ
Uh
2
finite elements in fluidsstabilizing techniques, conv-diff computational fluid dynamics I
• Let us follow the explanation of the stabilizing techniques by extrapolating theformulation to the two-dimensional steady-state convection diffusion equationwhich can be written as
• The Peclet number is now the vector
• Following the analogy with the balancing diffusion, the convection is only activein the direction of the resultant velocity U, and therefore the so-introducedbalancing diffusion should be only different from zero in the direction of thevelocity resultant
finite elements in fluidsstabilizing techniques, conv-diff computational fluid dynamics I
0
yk
yxk
xyV
xU
kh
2UPe
• This can be accomplish by considering weighting functions
rather than the previously used
where now is defined as
finite elements in fluidsstabilizing techniques, conv-diff computational fluid dynamics I
kh
Pe2U
i
kik
kkkkkk dx
dNUUhN
dydNV
dxdNU
UhNwNw
21
2*
dxdN
UUhNwNN k
kkkk 2 *
PePe 1coth
iiUUVU 22U
• This is equivalent to using a balancing diffusion to be used in the term
equal to
that therefore acts only in the direction of the velocity
• The length h can be taken as the maximum size in the direction of the velocityvector as shown in the picture
finite elements in fluidsstabilizing techniques, conv-diff computational fluid dynamics I
ijk~
dxN
xNk
jiij
U2~ ji
ij
UhUk
UU
hh
• Extending the GLS to the multidimensional problems where the formulationpresented was
• It is obtained
which is an identical stabilizing term to that of the streamline Petrov-Galerkinformulation
• The use of one or another approach is in most of the cases a matter of taste
finite elements in fluidsstabilizing techniques, conv-diff computational fluid dynamics I
dxQLdx
dNUhU
NL k
k
ˆ
0 2
dQ
xk
xxU
dxdNhUN
jjjj
i
kik
ˆˆ
U2
• Let us extend the GLS particulars to the vector valued Navier-Stokes equations
• The basis of the GLS stabilization technique is to add a term to both continuityand dynamic equations. These terms depend on the residual of the momentumeq., and therefore ensure the consistency of the stabilized formulation
• Se obtiene el mínimo de la suma de los cuadrados de los residuos. Para laformulación de Stokes, resulta
El extremo del funcional viene dado por
para cualquier valor de w y q, esto es:
dpppLs fufu,u 22
21
00
dqpdLs ,wu
Q x 022 Vw,qdpq
fuw
finite elements in fluidsstabilizing techniques, glscomputational fluid dynamics I
• Or eqivalently:
if we now add the ‘stabilizing’ terms multiplied by a certain factor to thedynamic and continuity equations, it is obtained
Vdp
wfuw 022
Qqdpq 02
fu
finite elements in fluidsstabilizing techniques, glscomputational fluid dynamics I
21 2
1
dwtdfwdfpuwdpwduwh
h
el
hh
hi
hi
hi
hi
n
e
hi
hi
hjji
hjjie
hhii
hji
hji ,,,,,,
h
el
dfpuqduqn
n
hi
hi
hjji
hie
hii
h 01
,,,,
• If linear interpolation functions for velocity and pressure are used, the terms withsecond derivatives vanish and we obtained
The terms lead to the appearance of non zero diagonal terms thatallow for the stabilization. As a consequence, basic elements with an equal orderinterpolation for velocity and pressure are stable (for example Q1Q1)
• The factor e is the stabilization parameter
where he is a measure of the element size, 0 can be taken as 1/3 for linearelements
dpq hi
hi ,,
finite elements in fluidsstabilizing techniques, glscomputational fluid dynamics I
2,,,2
1
dwtdfwdpwduwh
hhh
hi
hi
hi
hi
hhii
hji
hji
h
elel
dfqdpqduqn
n
hi
hie
n
n
hi
hie
hii
h
11,,,,
4
2
0e
eh
• Extrapolating to vector valued equations the issues regarded for the convectiondiffusion eq., the SUPG method can be also used for the Navier-Stokesequations
• The Galerkin formulation leads to a central approximation of the convective termand it is not optimal for convection dominate flows, that is for big Reynoldsnumbers as the stiffness matrix becomes more non-simmetric
• The SUPG formulation takes weighting functions as
where the upwind contribution to the weighting function p, is defined as
finite elements in fluidsstabilizing techniques, SUPGcomputational fluid dynamics I
iii pww
uˆ i
iuu iiuu2uh
hji
hjh
i
wukp
uˆ ,
• The multimensional definition of k is given by:
finite elements in fluidsstabilizing techniques, SUPGcomputational fluid dynamics I
2 huhu
khh
1coth
1coth
2hu h
2
hu h
heii
h ueu heii
h ueu
x1
x2
e
e
h
h
• Provided known initial conditions in the domain the steadysolution of the flow problems should be accurately transported in time
• The easiest way of integrating the unsteady equations is to consider a finitedifference approach for the local acceleration term, that is taking
• The forward differencing from level tn to level tn+1 would lead to the explicit (so-called Euler method) scheme
where the convective matrix is known at each iteration.
The accuracy of the former scheme is O(t and the step t must be restrictedfor stability
finite elements in fluidstime integration, classical time&spacecomputational fluid dynamics I
tOtt
nn
uuu 1
nnnnnn p
tFBAuucuuM
1
0uB nT
xu,xu 00 t
e
dNwM jiij
• An implicit scheme could also be obtained by backward differencing from tn+1
to tn , to obtain
where the non-linearities can be solved by either taking
that is
or taking
that anyway requires the non-linear terms to be small and to change slowly withrespect to time (t small)
finite elements in fluidstime integration, classical time&spacecomputational fluid dynamics I
11111
nnnnnn p
tFBAuucuuM
0uB 1nT
nn ucuc 1
dxhn
hn
hn 11 · uuwuc
nnnnnn p
tuCFBAuuuM
1111
• These approaches can be regarded in a joint way by considering the parameter in terms of which the equations could be written as
for the Euler method is obtained and for the formulation results into thebackward Euler algorithm
• For the so-called Crack-Nicolson formulation is obtained, which provideswith a second order accuracy as it leads to central differencing
finite elements in fluidstime integration, classical time&spacecomputational fluid dynamics I
0111111
nnnnnnnnnn pp
tFBAuucFBAuucuuM
0uuB nnT 11
• The splitting techniques would be also an alternative. For example the operatorcontributions associated with the non-linearities and the incompressibility can besplit to provide
and a second step that makes
formulation with many different variants
finite elements in fluidstime integration, classical time&spacecomputational fluid dynamics I
nnnnnn p
tucFBAu
/uuM ///
/
21212121
2
0uB 2/1nT
21111211
2 // BFAuuc
/uuM
nnnnnn p
t
• Time and space are linked in such a way that the discretization of one has aninfluence on the discretization of the other.
• An accurate spatial representation can be spoilt when it is transported in time ifthe integration algorithm is not able to propagate the information along thedirections prescribed by the convection problem
• This could be circumvented by resorting to a Lagrangian formulation, in whichthe convective terms disappear from the governing equations
• Nevertheless, the lagrangian description is not practicable as it would lead toexcessive distortion of the computational mesh
• The semi-Lagrangian, Lagrange-galerkin and characteristic-based methodsmake use the benefits to which the lagrangian description may lead to in theEulerian approach. They all take advantage to the fact that the unknown isconstant along a particle path or characteristic
finite elements in fluidstime integration, characteristicscomputational fluid dynamics I
• The classical first an second order time-stepping algorithms are not optimalfor convection-dominated problems as they are unable to take into account thedirectional character of the propagation of the information in the convection.
• This gets worst as the time stepping is increased
• That is why the higher order time-stepping schemes (such as the Taylor-Galerkin) are used allowing for un indirect propagation of the information alongthe characteristics
finite elements in fluidstime integration, characteristicscomputational fluid dynamics I
• Let us consider the one dimensional PDE of the convective-difussive transportequation
where although U and Q could depend on x and t, will be taken as constants
• We are going to transform the PDE into an ODE along the appropriate direction
• If we state the change of coordinates
the change of coordinates will give
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
0
Q
xk
xxU
t
Utx Utx
11UU
xxtt
x
t
• Taking Q=k=0, the convective transport equation
becomes
this is
the convective term disappears and the Galerkin formulation is available
• Therefore, only depends on that is
this means that the solution at point x and time t is equal o the solution at time t-t and point x-Ut
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
0 xt U
02 U
0
Utx
UtxtUUttUxtttUxtx ,,
• The concept of transport along the characteristic shows how an initial distribution
follows a uniform transport along the characteristic curve (line in this case) andremains constant along x=Ut+cte, as shown in the picture
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
xx 00,
x=Ut+cte
(x1,0)
(x1,0)=0(x1)
(x,t)
x
t
tt(x-Ut,t-t)
(x-Ut,t-t)=0(x1)
(x,t)=0(x1)
t
• The characteristic lines can beexpressed in their parametricexpression as
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
rxx
rtt
0
xU
t
x
t 3.53.5 4.5
• The idea is therefore to transform the PDE of the convective transport into anODE in which the unknown is integrated along the characteristic curves in the
space-time plane
• The variation of the unknown along the characteristic curve is given by
comparing this material derivative with the convective transport equation
we would have then
• From the first of the equations with t(r=0)=0, it is obtained
• The second equation gives for a constant U
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
0
drdx
xdrdt
tdrd
0
xU
t
Udrdx
1drdt
0drd
rt
Udtdx
drdx
0xUtx
• The equation
gives the characteristic curves along which the unknown verifies
this is, is a constant. In other words, we have transformed the PDE into anODE that has been integrated along the characteristic
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
0xUtx
0drd
Utxxtx 0000,
x-Ut=x0
(x,0)=0(x0)
x
t• The characteristics curves are a family ofequations in the x-t plane with parameterx0
• That is, for each value of x0 a differentcurve is obtained, where x0 is given by theinitial condition
0xUtx
• Example 1. Integrate
with the initial condition
Comparing the PDE with the material derivative of we have
that is
the initial condition is given by
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
021
xt
xx sin0,
21
dtdx
drdx1
drdt
0drd
02xtx 02
xtx
2sinsin, 0 txxtx
Their partial derivatives being
• Example 2. Integrate
with the initial condition
the equations to be verified now are
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
q.e.d. 02/cos
212/cos21
2/cos
2/cos21
txtxtx
x
txt
xxt
t2
xx 0,
xtdtdx
drdx 21
drdt
dtd
drd
The former equations give
from the total derivative we have
therefore, the solution to the PDE becomes
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
tdtx
dx 2 ktx 2ln2tcex 2
0texx
2
0txex
dtd
drd
dtd
kt ln tce 0
00, cextx tex0
22
, tttt xeexetx
Deriving the former expressions
• Example 3. Integrate
with the initial condition
the equations to be verified now are
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
q.e.d. 221
2122
2
2
tttt
tt
tt
xteetxe
x
etxt
02
xt
xx 0,
2dtdx
drdx1
drdt
0dtd
drd
If d/dr=0, then is constant along the characteristic and the equation dx/dt=2
gives
the initial condition gives
with derivatives
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
02 xtx 0
2 xtx
txx 20
tx 22
t
xtx
1
,
q.e.d. 0
1211
212
1
12 23
23
txtxtx
txx
txxt
• In the former examples the PDE has been solved by solving a ODE along thecharacteristic curve which has been obtained through another ODE
• This equations, anyway, not always can be solved analytically and anumerical solution has to be introduced in the characteristic approach thatsubdivides both the pressure and time spaces into discrete bits
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
• The convective transport equation
can be resolved by characteristics (that is the convective term eliminated) byusing a Lagrangian viewpoint as follows
• For a given space-time point (x1,t1), the characteristic line X passing through thepoint can be obtained from the differential equation
that verifies
with reference to the Lagrangian point of view, can be interpretedas providing the position at time t of a fluid particle transported by the convectionvelocity field U, which occupies the spatial position x1 at time t1. That is, definesthe trajectory of the particle
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
txQx
txUt
,,
ttxXUdt
ttxdX ;,;,11
11
1111 ;, xttxX
ttxXX ;, 11
• The value of the unknown along the characteristic will be
that is, its characteristic form. The solution of which will be
or
depending on the characteristic hitting the abscise or the ordinate axes
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
tQdtd
dttQtXtxt
tD
1,, 11
dttQXtxt
1
0011 0,
• The most obvious of the application of characteristics to the FEM would be toupdate the position of the mesh points in a Lagrangian way
we would have
for a constant velocity
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
11 n
n
t
t
ni
ni Udtxx
tUxx ni
ni 1
tn
tn+1= tn +t
x
t
xin xi
n+1
• On the updated mesh only the time-dependent diffusion has to be resolved(featured by symmetric matrices), but the process of continuously updating themesh would be unaffordable because of too large distortions in the mesh leadingto tough difficulties at the boundary
• After a single step, a return to the original one should be carried out byinterpolating from the updated values to the original mesh points
• The diffusion part of the computation will be carried out either on theoriginal or the final mesh leading to a certain splitting of the convectionand diffusion
• The general process will be generalized in the so-called characteristic-Galerkin methods
finite elements in fluidstime integration, the method of charac.computational fluid dynamics I
• The unknown is going to be split into a convective and a diffusive part
leading to the equations
computational fluid dynamics I
finite elements in fluidstime integration, characteristic-galerkin
***
0*
xU
t
0**
Q
xk
xt
• The convective diffusive transport can be written along the characteristic as
• In the former equation the convective term vanishes and the source and diffusionterms are average quantities along the characteristic. The equation is now self-adjoint and the Galerkin spatial approximation is optimal
• The time discretization of the former equation along the characteristic gives
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
0,
XQX
kXt
ttX
x
nn
x
nn Qx
kx
Qx
kxt
111
1
where is equal to zero for explicit formsand between zero and one for semi andfully implicit forms
• But this formulation requires meshupdating and leads to boundary problems x
n1
nn((x-)
=Ut
t
• The Taylor series expansion
or
would lead in this case to
that could also be used to take (with
where =Ut is the distance travelled by the particle in the x-direction
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
2
22
2 xx
nnn
x
n
nn
x xk
xxxk
xxk
x
221
21
xQQQ
nn
x
222
1
···!
2
2axafaxafafxf
···!
2
2xxfxxfxfxxf
• And U is an averaged value of the velocity along the characteristic and can betaken as
• Writing and substituting the former equations in the first expression it is obtained
• Substituting =Ut in the former expression and neglecting higher order terms, itis obtained
where
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
1
2
221
21
21
nnnnn
Qx
kxxxtt
n
xQQ
xk
xxk
x
2
2
21
xUtUUU
nnn
2121
1 //
nnn
nn Qx
kxx
Ut n
xQUt
xk
xUt
xU
xtt
222 2
22
nnn
xk
xxk
xxk
x
21
21 12/1
2
12/1
nnn QQQ
• The so-obtained formulation is identical to that obtained for the Taylor-Galerkinapproach (to be considered later).
• If we write the multidimensional problem in the fully explicit form, that isapproximating the n+1/2 in terms of n, we obtain
where the additional terms add the stabilizing diffusion in the streamline direction
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
n
iijj
nn Qx
kxx
Ut 1
n
ii
iikk
jji
i xQUt
xk
xxUt
xUU
xtt
222
• An alternative approximation for U would be taking it as
using the Taylor expansion
taking and substituting in the first equation it is obtained
where
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
2
1
x
nn UUU
xUtUUU
nnn
x
n
2
22/12/12/11
221
xUUt
xxUUt
xU
tnn
nnn
nnnn
xQUtQ
xk
xxUt
xk
xn
nn
n
2/12/12/1
22
2
12/1
nnn UUU
• As carried out before, we can approach
obtaining
• For multidimensional problems the formulation to be obtained is
(*)
which is very similar to the one obtained for the other approximation of U andidentical for the particular case U constant. This formulation will be the one to beused in what follows
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
nn UU 2/1
n
nnn Qx
kxx
Ut 1n
nn Qx
kxx
Ux
Ut
2
2
n
iij
jnn Qx
kxx
Ut
1 n
iij
j
k
nk Q
xk
xxU
xUt
2
2
• Writing the unknown in terms of the approximation
it is obtained
where
• The new terms introduced by the discretization along the characteristics afterintegrating by parts and dropping the second order derivatives (those relatedto diffusion) are
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
~N
ns
nu
nnnnn tt fKfKCM ~~~~~ 21
dTNNM
dUx i
i
TNNC
dx
kx ii
T NNK
..tbdQT
Nf
dUx
Ux i
i
Ti
iu NNK
21 ..
21 tbdQU
xT
ii
s
Nf
• The former formulation is only conditionally stable. For one dimensionalproblems, the stability condition for linear elements is given by
• In two dimensional problems the critical time step may given as
where t is given by the former equation and th2/2k is the diffusive limit forthe critical one-dimensional time step
• With t tcrit the steady state solution results in an almost identicalapproach to that obtained by using the optimal streamline upwindingprocedure. Therefore is the steady state conditions are pursued the time stepshould be t tcrit
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
Uhtt crit
tttttcrit
• The formulation for the steady state would be that in which
and the characteristic based formulation results in
which taking
results in a formulation identical to that of the Petrov-Galerkin formulationdropping the second order terms related to diffusion)
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
~~~ 1 nn
0~ su tt ffKKC
2UhUt
020
dxQL
dxdN
UhU
NL k
k ˆ
• If we carry out the same approach for vector valued functions such as the twodimensional field in the dynamic Navier-Stokes equation, the formulationobtained is
where
finite elements in fluidstime integration, characteristic-galerkincomputational fluid dynamics I
n
unn t
1 ~~~ fKKCM
dTNNM
dx
Ui
i
T NNC
dx
kx j
iji
T NNK
..2
tbdQx
Ut
i
T
iT
NNf
dx
tUUx j
jii
T
uNNK
2
• The Taylor–Galerkin is another approach to the same problem that leads to thesame formulation
• Starting with the Taylor series expansion in time (instead of space as before) of we would have
• If we write the convective-diffusive transport equation as
and derive it with respect to time
finite elements in fluidstime integration, Taylor-galerkincomputational fluid dynamics I
nn
Qx
kxx
Ut
32
221
2tO
tt
tt
nnnn
nn
Qx
kxx
Utt
2
2
• Now, we could substitute both former eqs. into the Taylor expansion to yield
• Assuming U and k constants
• Substituting in the former equation and neglecting higherorder terms, it is obtained
computational fluid dynamics I
nnnn Q
xk
xxU
ttQ
xk
xxUt
2
21
nnnn Q
tk
xtU
xtQ
xk
xxUt
2
21
nn
Qx
kxx
Ut
finite elements in fluidstime integration, Taylor-galerkin
n
nn Qx
kxx
Ut 1
n
QQx
kxx
Ukx
Qx
kxx
UUx
t
2
2
• Or also
which is the same one as the obtained for the Characteristic-Galerkin formulationfor constant U and k
• The multidimensional formulation turns out to be
computational fluid dynamics I
n
iijj
nn Qx
kxx
Ut 1
n
ijj
ij
jii
QUx
kx
Ux
UUx
t
2
2
finite elements in fluidstime integration, Taylor-galerkin
n
nn Qx
kxx
Ut 1
n
UQx
kx
Ux
Ux
t
2
2
2
• The so-called characteristics based split method is a procedure that, based in thesplitting methods developed by Chorin in the sixties, were introduced byZienkiewicz and Codina in the nineties
• The dynamic equation can be discretized in time using a characteristic Galerkinprocess.
• The dynamic equation is equal to the convection diffusion equation except for thepressure term, that can be treated as a known quantity provided that we haveanother way of evaluating the pressure
• Writing the Navier-Stokes equations as
where
finite elements in fluidstime integration, CBScomputational fluid dynamics I
2
n
iij
ijij
j
i Qgx
Uuxt
U
i
j
j
iij x
uxu
forcesbody igi
i
xU
tp
ct
2
1
sound of speedc
• In the former equation the source term Q taking as a known value equals
with
or with
• Using equation (*) from the characteristics method in which we have substituted by Ui it is obtained
(*)
finite elements in fluidstime integration, CBScomputational fluid dynamics I
i
nni x
pQ
22
i
n
i
n
i
n
xp
xp
xp
2
1
2 12
ii
n
i
n
xp
xp
xp
2
2
nn ppp 1
n
ini
j
nij
j
nijn
in
ii gQxx
UutUUU
21 n
iij
ij
k
nk gQ
xUu
xut
2
2
n
iij
jnn Qx
kxx
Ut
1 n
iij
j
k
nk Q
xk
xxU
xUt
2
2
• Let us introduce now the spilt in which we substitute a suitable approximation forQ which allows the calculations to proceed before the pn+1 is evaluated
• Let us introduce an auxiliary variable U*i such that the pressure gradient terms
are removed from the equation
(1)
• The former equation can then be solved by an explicit time step. Once thesolution U*
i is obtained we could compute
(2)
as far as we knew the pressure increment
finite elements in fluidstime integration, CBScomputational fluid dynamics I
n
ij
ij
kki
j
ij
j
ijniii g
xUu
xutg
xxUu
tUUU
2
**
k
ni
ki
n
ini
nii x
Qutx
ptUUUU
2
2*1
2
• From the dynamic equation for general incompressible fluids we obtain theequality
• Replacing Uin+1 by the known intermediate auxiliary variable Ui
*, rearranging andneglecting higher order terms it is obtained
(3)
where Ui* and pressure terms come from equation (2). This equation is self-
adjoint and a standard Galerkin formulation can be used
computational fluid dynamics I
i
i
i
ni
i
ni
n
xU
xUt
xUtp
c 12
11
iiii
n
i
i
i
ni
n
xxp
xxpt
xU
xUtp
c
2
2
2
112
1 *
finite elements in fluidstime integration, CBS
• Therefore, the equations to be solved are– Solve (1) for Ui
*
– Solve (3) for p
– Solve (2) for Ui thus stablishing the values at tn+1
computational fluid dynamics I
finite elements in fluidstime integration, CBS
• Some examples
finite elements in fluidssome examplescomputational fluid dynamics I
X
Y
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Detalle agitación. Situación Actual 27 Oct 2002 NNDetalle agitación. Situación Actual 27 Oct 2002 NN
computational fluid dynamics I
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finite elements in fluidssome examples
computational fluid dynamics I
Baffleplate
Sluge hoppers
Overflowlaunder
Baffle plateFoam sweeperInfluent intake
Walkway
Baffle slab
Sluddge scrappers
Sludge hopper
Influent pipe Sludge removal pipe
Effluentpipe
Foam sweepers
Foamlaunder
Overflowlaunder
Foam launder
Sluge scraper
Sluge withdrawal
Influent intake
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finite elements in fluidssome examples
computational fluid dynamics I
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Fram e 001 30 Jun 2 000 IT ER AC= 1Visc= 1 .0000000 00 00 00 00E -002Fram e 001 30 Jun 2 000 IT ER AC= 1Visc= 1 .0000000 00 00 00 00E -002
finite elements in fluidssome examples
computational fluid dynamics I
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finite elements in fluidssome examples
computational fluid dynamics I
finite elements in fluidssome examples
computational fluid dynamics I
finite elements in fluidssome examples
computational fluid dynamics I
finite elements in fluidssome examples
computational fluid dynamics I
finite elements in fluidssome examples
computational fluid dynamics I
finite elements in fluids