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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 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 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)
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