Introduction to Computational Fluid Dynamics

Fluid Mechanics and Heat Transfer. Basic equations. Continuity (Mass Conservation)    divv  0  div v  0 t incompressible

Momentum Equation (Navier-Stokes equations) Incompressible viscous fluid (  constant,   constant )    v         v v    F  grad p   v  t 

Lecture 2 –Basic equations. Dimensionless parameters 1 Introduction to Computational Fluid Dynamics

Energy equation dE d  1     p     divk grad T  dt dt    where E is the internal energy of a fluid particle (viscous compressible fluid)

- gas heating due to the compression d  1    p     p divv dt    continuity equation - mechanical work of the viscous forces transformation in heat

Lecture 2 –Basic equations. Dimensionless parameters 2 Introduction to Computational Fluid Dynamics

p By taking into account that the enthalpy of a unit of mass is i  E  we  obtain di dp     divk grad T  dt dt But for a perfect gas we have

di  c pdT and energy equation becomes d dp  c pT     divk grad T  dt dt where df Df  f      v  f dt Dt t is the substantial derivative.

Lecture 2 –Basic equations. Dimensionless parameters 3 Introduction to Computational Fluid Dynamics

Momentum Equation (for fluid saturated porous media) By a porous medium we mean a material consisting of a solid matrix with an interconnected void.

Porous media are present almost always in the surrounding medium, very few materials excepting fluids being non-porous.

heat exchanger processor cooler porous rock

Lecture 2 –Basic equations. Dimensionless parameters 4 Introduction to Computational Fluid Dynamics

Darcy’s (law) equation (momentum equation)  K  v   grad p   g  where

 v is the average velocity (filtration velocity, superficial velocity, seepage velocity or Darcian velocity)

 K  k g is the permeability of the porous medium

Darcy’ law expresses a linear dependence between the pressure gradient and the filtration velocity. It was reported that this linear law is not valid for large values of the pressure gradient.

Lecture 2 –Basic equations. Dimensionless parameters 5 Introduction to Computational Fluid Dynamics

Darcy’s law extensions Forchheimer’s extension

  c   p   v  F  v v K K where cF is a dimensionless parameter depending on the porous medium

Brinkman’s extension    p   v  ~  v K where ~ is an effective viscosity

Brinkman-Forchheimer’s extension   c    p   v  F  v v  ~  v K K

Lecture 2 –Basic equations. Dimensionless parameters 6 Introduction to Computational Fluid Dynamics

Energy equation for porous media  T   c   c v  T   k  T  p m  p fluid  e  t where

 c    c  1  c  p m  p fluid  p solid

ke   k fluid  1ksolid

Lecture 2 –Basic equations. Dimensionless parameters 7 Introduction to Computational Fluid Dynamics

Bouyancy driven flow1

 There exists a large class of fluid flows in which the motion is caused by buoyancy in the fluid.  Buoyancy is the force experienced in a fluid due to a variation of density in the presence of a gravitational field. According to the definition of an incompressible fluid, variations in the density normally mean that the fluid is compressible, rather than incompressible.  For many of the fluid flows of the type mentioned above, the density variation is important only in the body-force term of the momentum equations. In all other places in which the density appears in the governing equations, the variation of density leads to an insignificant effect.  Buoyancy results in a force acting on the fluid, and the fluid would accelerate continuously if it were not for the existence of the viscous forces.  The situation depicted above occurs in natural convection.

1 I.G. Currie, Fundamental Mechanics of Fluids, 3rd ed., Marcel Dekker, New York, 2003

Lecture 2 –Basic equations. Dimensionless parameters 8 Introduction to Computational Fluid Dynamics

Generally, density is a function of temperature and concentration

  T,c

Black Sea salinity, temperature and density (Ecological Modelling, 221, 2010, p. 2287-2301)

Lecture 2 –Basic equations. Dimensionless parameters 9 Introduction to Computational Fluid Dynamics

Boussinesq’s approximation

, Momentum

Lecture 2 –Basic equations. Dimensionless parameters 10 Introduction to Computational Fluid Dynamics

Dimensionless equations u v w    0 x y z

 u u u u  p  2u 2u 2u    u  v  w   F         x  2 2 2   t x y z  x  x y z 

 v v v v  p  2v 2v 2v    u  v  w   F         y  2 2 2   t x y z  y  x y z 

 w w w w p  2w 2w 2w   u  v  w   F         z  2 2 2   t x y z  z  x y z 

 T T T T   2T 2T 2T  c   u  v  w   k      q( 0) p  2 2 2   t x y z   x y z 

2 2 2 2 2 2   u   v   w  v u   w v   u w    2   2   2               x x x x y y z z x              

Lecture 2 –Basic equations. Dimensionless parameters 11 Introduction to Computational Fluid Dynamics

We introduce further the dimensionless variables  t x y z u v w  F p T  T   , X  , Y  , Z  , U  , V  , W  , F ' , P  ,   0 t0 L L L U0 U0 U0 g p0 T into the governing equations

u   U0 U u  X U0 U  UU0   ,  UU0   , t  t t0  x X x L X p  X p P  P p   0 , x X 0 x L X

2u   u   U U  U 2U X U 2U      0   0  0 x2 x  x  x  L X  L X 2 x L2 X 2 T   T  T  X T     T  T0   ,    T  T0   t  t t0  x X x L X

2T   T    T   T 2 X T 2U         x2 x  x  x  L X  L X 2 x L2 X 2

Lecture 2 –Basic equations. Dimensionless parameters 12 Introduction to Computational Fluid Dynamics and we obtain the dimensionless equations

U V W    0 X Y Z

L U U V W gL p P   2U 2U 2U  U V W  F '  0      2 x 2  2 2 2  t0U0  X Y Z U0  U0 X U0L  X Y Z 

 k       2 2 2 L    Q  c p         U U V W         0   2 2 2  t0U0  X Y Z U0L   X Z Z   c p LT

Lecture 2 –Basic equations. Dimensionless parameters 13 Introduction to Computational Fluid Dynamics

Dimensionless numbers A. Strouhal number The Strouhal number (St) is a dimensionless number describing oscillating flow mechanisms: L L oscillation St    t0U0 U0 average velocity

B. Froude number The Froude number (Fr) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). In naval architecture the Froude number is a very significant figure used to determine the resistance of a partially submerged object moving through water, and permits the comparison of similar objects of different sizes, because the wave pattern generated is similar at the same Froude number only. U Fr  0 gL

Lecture 2 –Basic equations. Dimensionless parameters 14 Introduction to Computational Fluid Dynamics

C. Euler number It expresses the relationship between a local pressure drop e.g. over a restriction and the kinetic energy per volume, and is used to characterize losses in the flow, where a perfect frictionless flow corresponds to an Euler number of 1. p 0 Eu  2 U0 2 Usually, we take p0  U0 and we obtain Eu = 1.

D. Reynolds number U L U L Re  0  0   The Reynolds number is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.  laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;  turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

Lecture 2 –Basic equations. Dimensionless parameters 15 Introduction to Computational Fluid Dynamics

E. Prandtl number

c   /   Pr  p   k k / c p  It is the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number contains no such length scale in its definition and is dependent only on the fluid and the fluid state. As such, the Prandtl number is often found in property tables alongside other properties such as viscosity and thermal conductivity. •gases - Pr ranges 0.7 - 1.0 •water - Pr ranges 1 - 10 •liquid metals - Pr ranges 0.001 - 0.03 •oils - Pr ranges 50 - 2000

E. Eckert number It expresses the relationship between a flow's kinetic energy and the boundary layer enthalpy difference, and is used to characterize heat dissipation.

U 2 Εc  0 c p T

Lecture 2 –Basic equations. Dimensionless parameters 16 Introduction to Computational Fluid Dynamics

Using the above dimensionless numbers we obtain: U V W    0 X Y Z U U V W 1 P 1  2U 2U 2U  St U V W  F ' Eu      x  2 2 2   X Y Z Fr X Re  X Y Z  ….    Q 1   2  2  2  Ec St U V W         2 2 2   X Y Z Re Pr  X Z Z  Re or using differential operators    V  0 (23) V   1  1  St  V V  F'Eu P  V (24)  Fr Re   1 Ec St  V      (25)  Re Pr Re

Lecture 2 –Basic equations. Dimensionless parameters 17 Introduction to Computational Fluid Dynamics

Comments  Dimensionless numbers allow for comparisons between very different systems and tell you how the system will behave  Many useful relationships exist between dimensionless numbers that tell you how specific things influence the system  When you need to solve a problem numerically, dimensionless groups help you to scale your problem.  Analytical studies can be performed for limiting values of the dimensionless parameters

Express the governing equations in dimensionless form to: (1) identify the governing parameters (2) plan experiments (3) guide in the presentation of experimental results and theoretical solutions

If the flow is not oscillatory, we take usually t0  L U0 and thus St =1. If t0   we have St = 0 and the flow is steady.

Lecture 2 –Basic equations. Dimensionless parameters 18 Introduction to Computational Fluid Dynamics

Numerical methods for initial values problems (IVP or Cauchy problem) Pendulum Mathematical model:

d 2 g d L  sin  0,  (t0 )   0 , (t0 )   '0 dt2 L dt



Radioactive elements decay:

dm   m, m(t )  m dt 0 0 where m is the mass of the radioactive element and  is the decay constant.

Lecture 2 –Basic equations. Dimensionless parameters 19 Introduction to Computational Fluid Dynamics

Theoretical aspects

The general Cauchy problem:  y'(t)  f (t, y), a  t  b  y(a)  ya Using numerical methods we obtain a discrete approximation of y in some points, called nodes, which form a grid (mesh).

A grid for the interval [a,b] having a constant step h is: a,a  h,a  2h,...,a  ih,...,a  (N 1)h,b or

ti  a  (i 1)h, i  1,..., N where N is the number of subintervals, and the step is constant: h  (b  a)/(N 1).

Lecture 2 –Basic equations. Dimensionless parameters 20 Introduction to Computational Fluid Dynamics

By using a numerical method we obtain the approximated discrete values not y(ti )  yi , i  1,..., N

y(t)

yi+2 yi+1

yi-1 yi

yi-2 t i+1 i-2 i-1 i i+2

Lecture 2 –Basic equations. Dimensionless parameters 21 Introduction to Computational Fluid Dynamics

One step methods An example: The Euler method (Leonhard Euler-(1707 – 1783))

 y'(t)  f (t, y), a  t  b  y(a)  ya By using a Taylor expansion (y should be of class C 2) we have: h2 y(t )  y(t  h)  y(t )  h y'(t )  y"( ), (t ,t ) i1 i i i 2 i i i i1 Thus, one get: h2 y(t )  y(t )  h f (t , y(t ))  y"( ) i1 i i i 2 i By dropping the last term the Euler formula is obtained: y  y 0 a yi1  yi  h f (ti , yi ), i 0,1,..., N 1

Remark: yi 1 depends on yi , ti and h.

Lecture 2 –Basic equations. Dimensionless parameters 22 Introduction to Computational Fluid Dynamics

(Richard L. Burden and J. Douglas Faires , Numerical Analysis,Ninth Edition, 2011).

Lecture 2 –Basic equations. Dimensionless parameters 23 Introduction to Computational Fluid Dynamics

One may notice that the Euler method follow the tangent for the current node to approximate the value in the next node.

Lecture 2 –Basic equations. Dimensionless parameters 24 Introduction to Computational Fluid Dynamics

Same problem for h = 0.2 (11 nodes). Comparison with the

Lecture 2 –Basic equations. Dimensionless parameters 25 Introduction to Computational Fluid Dynamics

An one step method for the Cauchy problem is given by y  y  h(t , y ,h), [t, y][a,b] Rn , h  0 i1 i i i  :[a,b] Rn1  Rn ~ Considering the exact solution (in the grid points) y(ti ) we define the local truncation error as follow: y  y ~y(t )  ~y(t ) ~y(t )  ~y(t ) T (t , y ,h)  i1 i  i1 i  (t , y ,h)  i1 i i i h h i i h (the difference between the approximation increment and the exact increment for one step) A method  is consistent if T(t, y,h)  0 uniformly when h  0 for (t, y)[a,b] Rn .

(it is necessary that (t, y,0)  f (t, y))

Lecture 2 –Basic equations. Dimensionless parameters 26 Introduction to Computational Fluid Dynamics

A method  is of order p if T (t, y,h)  Kh p

uniformly on [a,b] Rn , where is a vector norm, and K is constant. We may use the following notation: T (t, y,h)  O(h p ), h  0

(for p  1 the method is consistent). It can be shown that Euler method is a method of order one, O(h).

Lecture 2 –Basic equations. Dimensionless parameters 27 Introduction to Computational Fluid Dynamics

Improved Euler methods

evaluation of the tangent is made in an intermediary point of the interval ti ,ti1 

y(t)

slope = f(ti+1, yi+hf(ti, yi)) yexact ymodified yEuler y average slope i ti ti+1/2 ti+1 t slope = f(ti, yi)

Modified Euler

ti ti+1 Heun

Lecture 2 –Basic equations. Dimensionless parameters 28 Introduction to Computational Fluid Dynamics

not 1 -evaluation in the middle of t ,t , t  t  h i i1 i1/ 2 i 2    1 1  y(t )  y(t )  h f t , y   y(t )  h f t  h, y  hf (t , y ) i1 i i1/ 2 i1/ 2 i i 2 i 2 i i    Euler     1 1  y(ti1)  y(ti )  h f ti  h, yi  h f (ti , yi )  2 2   K (t ,y ) 1 i i  K2 (ti ,yi ,h) K (t , y )  f (t , y ) K (t , y )  f (t , y ) 1 i i i i 1 i i i i  1 1  K 2 (ti , yi ,h)  f ti  h, yi  hK1  K ( t , y ,h)  f t  h, y  hK  2 i i i 2 i 2 1   1 yi1  yi  hK1  K 2  yi 1  yi  hK2 2

modified Euler method Heun’s method (trapezoidal rule)

Lecture 2 –Basic equations. Dimensionless parameters 29 Introduction to Computational Fluid Dynamics

Runge-Kutta type methods

evaluation in intermediary points of ti ,ti1 

Standard Runge-Kutta method h y  y  K  2K  2K  K  i1 i 6 1 2 3 4

K1  f (ti , yi )  1 1  K 2  f ti  h, yi  hK1  2 2 

 1 1  K 3  f ti  h, yi  hK 2   2 2  K 4  f ti  h, yi  hK3  Carl David Tolmé Runge Martin Wilhelm Kutta (1856 – 1927) (1867 – 1944)

Lecture 2 –Basic equations. Dimensionless parameters 30 Introduction to Computational Fluid Dynamics

Matlab solvers for Cauchy problems (ODE) Syntax

[t,Y] = solver(odefun,tspan,y0,options, p1, p2, ...) or sol = solver(odefun,[t0 tf],y0...) where solver can be ode45, ode23, ode113, ode15s, ode23s, ode23t or ode23tb.

Input parameters (selection) odefun – right hand member of the Cauchy problem tspan –integration interval. y0– initial value options – solver options.

Lecture 2 –Basic equations. Dimensionless parameters 31 Introduction to Computational Fluid Dynamics

Output parameters: t – column vector of time points; y – solution array sol – solution structure

Lecture 2 –Basic equations. Dimensionless parameters 32 Introduction to Computational Fluid Dynamics

Lecture 2 –Basic equations. Dimensionless parameters 33 Introduction to Computational Fluid Dynamics

Example: Solve pendulum equation using ode45:

d 2 g d  sin  0,  (0)  0, (0)  0.1 dt2 L dt

We note y1   and rewrite the system in the form dy 1  y ; dt 2 dy g 2   sin y dt L 1 dy y 0  0; 1 0  0.1 1 dt

function dy=fPendul(t,y,flag,g,L) dy=[y(2);-g/L*sin(y(1))];

Lecture 2 –Basic equations. Dimensionless parameters 34 Introduction to Computational Fluid Dynamics

%pendulum equation a=0;b=pi/2;%integration interval g=9.8;%accelaration due to the gravity L=0.1;%length of the pendulum y0=[0,0.1];%initial conditions options=odeset('RelTol',1e-8);%modify the options %use the solver [t,y]=ode45('fPendul',[a,b],y0,options,g,L) plot(t,y(:,1));

Lecture 2 –Basic equations. Dimensionless parameters 35