DIFFERENTIAL FORMULATION OF THE BASIC LAWS

DIFFERENTIAL FORMULATION

OF THE BASIC LAWS

2.1 Introduction

x Differential formulation of basic laws: x Conservation of mass

x Conservation of momentum x Conservation of energy

2.2 Flow Generation

(i) Forced convection. Motion is driven by mechanical means. (ii) Free (natural) convection. Motion is driven by natural forces. 2.3 Laminar vs. Turbulent Flow

x Laminar flow: no fluctuations in velocity, pressure, temperature, … x Turbulent flow: random fluctuations in velocity, pressure, temperature, …

x Transition from laminar to turbulent flow: Determined by the Reynolds number: x Flow over a flat plate: Re_t V_fx_t/

ǎ

| 500,000

x Flow through tubes: Re_t= uD

v

| 2300

t turbulent u laminar u t

2.1

Fig.

2.4 Conservation of Mass: The Continuity Equation 2.4.1 Cartesian Coordinates

x The principle of conservation of mass is applied to an element dxdydz

(2.1)

Rate of mass added to element - Rate of mass removed from element =

Rate of mass change within element

dx dy x y ) (a _(b) 2.2 Fig. dy y m m_y y w w (G ) G y m G x m G dy y m m_x x w w (G ) G

x Expressing each term in terms of velocity components gives continuity equation

0 w w w w w w w w w z v y u x t U U U U (2.2a)

x This equation can be expressed in different forms:

0 » ¼ º « ¬ ª w w w w w w w w w w w w w w z w y v x u z w y v x u t U U U U U (2.2b) or 0 ˜ ’ V t D D & U U _(2.2c) or 0 ˜ ’ w w V t & U U _(2.2d)

x For constant density (incompressible fluid):

0 ˜

’ V& (2.3) 2.4.2 Cylindrical Coordinates

1

0 1 w w w w w w w w z r v z v r v r r r t U T U U U T (2.4) 2.4.3 Spherical Coordinates

0 sin 1 sin sin 1 1 ₂ 2 _w w w w w w w w I T U I T T U T T U U v r v r v r r r t r (2 .5)

2.5 Conservation of Momentum: The Navier-Stokes Equations of Motion 2.5.1 Cartesian Coordinates

x Application of Newton’s law of motion to the element shown in Fig. 2.5, gives

¦

GF& (Gm)a& (a) x Application of (a) in the x-direction, gives

(b)

¦

GFx (Gm)ax

x Each term in (b) is expressed in terms of flow

field variables: density, pressure, and velocity components:

dx dy x y z dz 2.5 Fig.

Mass of the element:

dz dy dx m U

G (c) Acceleration of the element a_x:

t u z u w y u v x u u Dt Du dt du a_x w w w w w w w w (d)

Substituting (c) and (d) into (b)

dxdydz Dt Du F_x

¦

G U

g dxdydz F_{x body x}

¦

G U (e) Forces:

(i) Body force

(g)

Summing up all the x-component forces shown in Fig. 2.6 gives

dxdydz z y x įFx _surface xx yx zx _¸¸ ¹ · ¨ ¨ © § w w w w w w

¦

V W W (h) Combining the above equations

x-direction: z y x g Dt Du _xx yx _zx x w w w w w w V W W U U (2.6a) By analogy: y-direction: z y x g Dt Dv xy yy zy y w w w w w w W V W U U (2.6b) z-direction: z y x g Dt Dw xz yz _zz z w w w w w w W W V U U (2.6c) IMPORTANT

x THE NORMAL AND SHEARING STRESSES ARE EXPRESSED IN TERMS OF VELOSICTY AND PRESSURE. THIS IS VALID FOR NEWTOINAN FLUIDS. (See equations 2.7a-2.7f).

x THE RESULTING EQUATIONS ARE KNOWN AS THE NAVIER-STOKES EQUAITONS OF MOTION

x SPECIAL SIMPLIFIED CASES: (i) Constant viscosity

V

V p

g Dt

V

D& & & & 2&

3 1 ’ ˜ ’ ’ ’ P P U U (2.9)

x (2.9) is valid for: (1) continuum, (2) Newtonian fluid, and (3) constant viscosity

(ii) Constant viscosity and density

V p

g Dt

V

D & & & 2&

’ ’ P U U (2.10)

x (2.10) is valid for: (1) continuum, (2) Newtonian fluid, (3) constant viscosity and (4) constant density.

x The three component of (2.10) are

x: ¸ ¸ ¹ · ¨ ¨ © § w w w w w w w w ¸¸ ¹ · ¨¨ © § w w w w w w w w 2 2 2 2 2 2 z u y u x u x p g z u w y u v x u u t u x P U U (2.10x)

y-¸ ¸ ¹ · ¨ ¨ © § w w w w w w w w ¸¸ ¹ · ¨¨ © § w w w w w w w w 2 2 2 2 2 2 z v y v x v y p g z v w y v v x v u t v y P U U (2.10y) z -¸ ¸ ¹ · ¨ ¨ © § w w w w w w w w ¸¸ ¹ · ¨¨ © § w w w w w w w w 2 2 2 2 2 2 z w y w x w z p g z w w y w v x w u t w z P U U (2.10z) 2.5.2 Cylindrical Coordinates

The three equations corresponding to (2.10) in cylindrical coordinates are (2.11r), (2.11T ), and (2.11z).

2.5.3 Spherical Coordinates

The three equations corresponding to (2.10) in spherical coordinates are (2.11r), (2.11T ), and (2.11I).

2.6 Conservation of Energy: The Energy Equation 2.6.1 Formulation

x The principle of conservation of energy is applied to an element dxdydz

dx dy x y z dz 2.7 Fig. convection

kinetic by transport energy gs surroundin on element by done work of rate Net conduction by addition heat of rate Net

_

and internal of rate Net (2.14) of change of Rate kinetic and internal element of energy

D

C

B

A

x The variables u, v, w, p, T, andUare used to express each term in (2.14).

x Assumptions: (1) continuum, (2) Newtonian fluid, and (3) negligible nuclear, electromagnetic and radiation energy transfer.

x Detailed formulation of the terms A, B, C and D is given in Appendix A x The following is the resulting equation

) P E U ’˜ ’ Dt Dp T T k Dt DT c_p (2.15) x (2.15) is referred to as the energy equation

p T»¼ º «¬ ª w w U U E 1 (2.16)

x The dissipation function ) is associated with energy dissipation due to friction. It is important in high speed flow and for very viscous fluids. In Cartesian coordinates ) is given by 2 3 2 2 2 2 2 2 2 2 ¸¸ ¹ · ¨¨ © § w w w w w w » » ¼ º « « ¬ ª ¸ ¹ · ¨ © § w w w w ¸¸ ¹ · ¨¨ © § w w w w ¸¸ ¹ · ¨¨ © § w w w w » » ¼ º « « ¬ ª ¸ ¹ · ¨ © § w w ¸¸ ¹ · ¨¨ © § w w ¸ ¹ · ¨ © § w w z w y v x u z u x w y w z v x v y u z w y v x u ) (2.17)

2.6.2 Simplified Form of the Energy Equation

x Cartesian Coordinates

(i) Incompressible fluid. Equation (2.15) becomes ) P U ’˜k’T Dt DT c_p (2.18)

(ii) Incompressible constant conductivity fluid. Equation (2.18) is simplified further if the conductivity k is assumed constant

) P U k’ T Dt DT c_p 2 (2.19a) or ) P U _¸¸ ¹ · ¨ ¨ © § w w w w w w ¸¸ ¹ · ¨¨ © § w w w w w w w w 2 2 2 2 2 2 z T y T x T k z T w y T v x T u t T c_p (2.19b)

(iii) Ideal gas. (2.15) becomes

) P U ’˜ ’ Dt Dp T k Dt DT c_p (2.22) or ) P U ’˜k’T p’˜V Dt DT c_v & (2.23)

x Cylindrical Coordinates. The corresponding energy equation in cylindrical coordinate is given in (2.24)

x Spherical Coordinates. The corresponding energy equation in cylindrical coordinate is given in (2.26)

2.7 Solutions to the Temperature Distribution

x The flow field (velocity distribution) is needed for the determination of the temperature distribution.

IMPORTANT:

x Table 2.1 shows that for constant density and viscosity, continuity and momentum (four equations) give the solution to u, v, w, and p. Thus for this condition the flow field and temperature fields are uncoupled (smallest rectangle).

x For compressible fluid the density is an added variable. Energy equation and the equation of state provide the fifth and sixth required equations. For this case the velocity and temperature fields are coupled and thus must be solved simultaneously (largest rectangle in Table 2.1).

Basic law _EquationsNo. of Unknowns

Viscosity relation Conductivity relation ) (p T k k , ) (p,T P P 1 T u v w p U P P P TABLE 2.1 U U U v 1 3 1 1 T T p p p w w v u u T 1 u v w U Continuity Momentum Energy State of Equation 1 T p k k

2.8 The Boussinesq Approximation

x Fluid motion in free convection is driven by buoyancy forces.

x Gravity and density change due to temperature change give rise to buoyancy.

x According to Table 2.1, continuity, momentum, energy and equation of state must be solved simultaneously for the 6 unknowns: u. v, w, p,Tand U

x Starting with the definition of coefficient of thermal expansionE, defined as

p T»¼ º «¬ ª w w U U E 1 (2.16)

or f f f | T T U U U E 1 (f)

This result gives

)

( _f

Uf EUf T T

U (2.28) x Based on the above approximation, the momentum equation becomes

T T

p p

V

g Dt

V

D & & 1 2&

’ ’ _f f f

v

U E (2.29) 2.9 Boundary Conditions

(1) No-slip condition. At the wall, y 0

0 ) , , 0 , (x z t V& (2.30a) or 0 ) , , 0 , ( ) , , 0 , ( ) , , 0 , (x z t v x z t w x z t u (2.30b) (2) Free stream condition. Far away from an object (y f)

f

f z t V

x

u( , , , ) (2.31) Similarly, uniform temperature far away from an object is expressed as

f

f z t T

x

T( , , , ) (2.32) (3) Surface thermal conditions. Two common surface thermal conditions are used in the analysis of convection problems. They are:

(i) Specified temperature. At the wall:

s T t z x T( ,0, , ) (2.33) (ii) Specified heat flux. Heated or cooled surface:

o q y t z x T k r cc w w ( ,0, , ) (2.34)

2.10 Non-dimensional Form of the Governing Equations: Dynamic and Thermal Similarity Parameters

x Express the governing equations in dimensionless form to: x (1) identify the governing parameters

x (2) plan experiments

x (3) guide in the presentation of experimental results and theoretical solutions x Dimensional form:

x Unknown variables are: u,v w, p, and These variables depend on the four independent variables. In addition various quantities affect the solutions. They are

.

T

x p_f,T_f,V_f, T_s, L g, , p_f,and x Fluid properties c_p, k,E,P, and U x Geometry

2.10.1 Dimensionless Variables

x Dependent and independent variables are made dimensionless as follows:

f V V V & &_* , 2 * ( ) f f f V p p p U , ( ) ) ( * f f T T T T T s , * , g g g & & (2.35) L x x* , L y y* , L z z* , t L V t* f

x Using (2.35) the governing equations are rewritten in dimensionless form.

2.10.2 Dimensionless Form of Continuity

0 * * ’˜V t D D & U U _(2.37) x No parameters appear in (2.37)

2.10.3 Dimensionless Form of the Navier-Stokes Equations of Motion

* 2 * * * * * 2 * * 1 V Re P g T Re Gr Dt V

D& & &

’ ’

(2.38)

x Constant (characteristic) quantities combine into two governing parameters:

v

L V L V Re{ f f P U

, Reynolds number (viscous effect) (2.39)

2 3

v

L T T g

Gr{ E s f , Grashof number (free convection effect) (2.40) 2.10.4 Dimensionless Form of the Energy Equation

x Consider two cases:

* * 2 * * * 1 ) Re Ǽc T RePr Dt DT ’ (2.41a)

x Constant (characteristic) quantities combine into two additional governing parameters: D U U P P

v

p p c k k c Pr / /

, Prandtl number (heat transfer effect) (2.42)

) ( 2 f f T T c V Ǽc s p

, Eckert number (dissipation effect – high speed, large viscosity) (2.43)

(ii) Ideal gas, constant conductivity and viscosity

* * * * 2 * * 1 ) Re Ǽc Dt Dp Ǽc T RePr Dt DT ’ (2.41b)

x No new parameters appear.

2.10.5 Significance of the Governing Parameters x Dimensionless temperature solution:

) , , , ; , , , ( * * * * * Ec Gr Pr Re t z y x f T (2.45) x NOTE:

x Six quantities: p_f,T_f, T_s, V_f, L g, and five properties k,E,P, and U , are replaced by four dimensionless parameters: Re,Pr,GrandEc.

, p c

x Special case: negligible free convection and dissipation: Two governing parameters: ) , ; , , , ( * * * * * Pr Re t z y x f T (2.46) x Geometrically similar bodies have the same solution when the parameters are the

same.

x Experiments and correlation of data are expressed in terms of parameters rather than dimensional quantities.

x Numerical solutions are expressed in terms of parameters rather than dimensional quantities.

2.10.6 Heat Transfer Coefficient: The Nusselt Number x Local Nusselt number:

(2.51) ) Ec Gr Pr Re x f Nu_x = ( *; , , , x Special case: negligible buoyancy and dissipation:

(2.52) ) Pr Re x f Nu_x = ( *; , x Free convection, negligible dissipation

(2. 53) ) Pr Gr x f Nu_x = ( *; ,

x For the average Nusselt number, x* is eliminated in the above. 2.9 Scale Analysis

x A procedure used to obtain order of magnitude estimates without solving governing equations.