Laminar Boundary Layer Theory

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					                                                                                            or
                                                                                                                                   Du           ∂           ∂          ∂
                                                                                                             Momentum, x : ρ           = ρgx +    (Πxx ) +    (Πxy ) + (Πxz )
                                                                                                                                   Dt          ∂x          ∂y         ∂z
                         Laminar Boundary Layer Theory                                                                             Dv          ∂           ∂          ∂
                                                                                                                Momentum, y : ρ        = ρgy +    (Πyx ) +    (Πyy ) + (Πyz )            (I 5)
                                                                                                                                   Dt          ∂x          ∂y         ∂z
                                                                                                                                  Dw            ∂           ∂          ∂
                                      Autumn Semester 2003                                                      Momentum, z : ρ        = ρgz +    (Πzx ) +    (Πzy ) + (Πzz )
                                                                                                                                   Dt          ∂x          ∂y         ∂z
                                                                                            [Π]: tensor of all surface stresses;
                                                                                            g = (gx , gy , gz ): acceleration of gravity (volume forces).
I        Basic equations for a viscous, compressible flow
The equations governing are derived using physical conservation laws:                                                y 6                ∂Πyy         6
                                                                                                                                Πyy +        δy
                                                                                                                                         ∂y               Πxy
     1. Conservation of mass                                                                                                                               -
                                                                                                                                 Πyx                                    ∂Πxx
     2. Conservation of momentum                                                                                                       ?
                                                                                                                                                                Πxx +        δx
                                                                                                                                          δy                       -    ∂x

                                                                                                                                 Πxx                 δx         6        ∂Πyx
     3. Conservation of energy.                                                                                                                                  Πyx +        δx
                                                                                                                                                                         ∂x
                                                                                                                                       ∂Πxy
                                                                                                                              Πxy +         δy         Πyy
I.1       Continuity equation (conservation of mass)
                                                                                                                                        ∂y           ?
                                                                                                                                                                            -
Local equation for conservation of mass:                                                                                                                                   x
                                              Dρ
                                                 +ρ       · v = 0,                  (I 1)
                                              Dt                                                             Figure 1: Surface stresses on a fluid element in 2 dimensions.
where ρ is the density of the fluid and v = (u, v, w) the velocity of a fluid element (in a   Π suffix notation: first suffix indicates reverse of direction in which stress acts, second suffix
Cartesian coordinate system (x, y, z)).                                                     indicates the face normal to axis direction on which stress acts.
D        ∂        ∂      ∂      ∂
Dt
     =   ∂t
              + u ∂x + v ∂y + w ∂z is the substantive (or material) derivative.
                                                                                                By taking moments about axes through the cube centre, it is possible to show that the
                                                                                            stress tensor is symmetric, i.e.
     Another form of the equation is (check as an exercise):
                                                                                                                         Πxy = Πyx ,     Πzx = Πxz ,           Πyz = Πzy                 (I 6)
                                 ∂ρ   ∂        ∂        ∂
                                    +   (ρu) +    (ρv) + (ρw) = 0                   (I 2)
                                 ∂t ∂x         ∂y       ∂z                                  I.2.2    Constitutive relations

     If the fluid is incompressible, ρ = constant so that                                    Stokes (1845) postulates:

                                                 Dρ                                              1. Linear relationship between stress and rate of strain (valid for Newtonian fluids),
                                                    = 0,
                                                 Dt                                              2. Isotropy, e.g. the fluid has no preferred direction in which the strain may be induced.
     hence, the continuity equation reduces to                                              The constitutive relations are:
                                                  · v = 0.                          (I 3)                                                       ∂u                  ∂u ∂v
                                                                                                                 Πxx = −p + λ       · v + 2µ       ,      Πxy = µ     +   ,
                                                                                                                                                ∂x                  ∂y ∂x
I.2       Conservation of momentum                                                                                                              ∂v                  ∂v ∂w
                                                                                                                 Πyy = −p + λ       · v + 2µ       ,      Πyz = µ      +    ,            (I 7)
I.2.1         The Cauchy equations                                                                                                              ∂y                  ∂z   ∂y
Newton’s second law yields the Cauchy equations,                                                                                               ∂w                   ∂u ∂w
                                                                                                                 Πzz = −p + λ       · v + 2µ      ,       Πxz = µ      +    ,
                                                                                                                                               ∂z                   ∂z   ∂x
                                              Dv
                                          ρ      = ρg +       · [Π]                 (I 4)   where µ is the first coefficient of viscosity (dynamic viscosity), λ is the second coefficient of
                                              Dt
                                                                                            viscosity (bulk viscosity) and p is the pressure.

                                                      1                                                                                          2
   The expression for [Π] can be substituted into the Cauchy equations to obtain the Navier–      or in Cartesian tensor notation,
Stokes equations for a compressible fluid. In doing so, remember that µ, λ and ρ all vary
with (x, y, z, t).
                                                                                                                                 ∂ui
                                                                                                                                     = 0                                              (I 14)
                                                                                                                                 ∂xi
I.3     Cartesian tensor notation                                                                                                Duj         ∂p      ∂ 2 uj
                                                                                                                               ρ     = ρgj −     +µ         ,                         (I 15)
Write                                                                                                                            Dt          ∂xj    ∂xi ∂xi

               (x, y, z) as (x1 , x2 , x3 );                                                      or equivalently,
               (u, v, w) as (u1 , u2 , u3 );
               (gx , gy , gz ) as (g1 , g2 , g3 ).
                                                                                                                                  ∂u ∂v ∂w
                                                                                                                                     +    +       = 0                                 (I 16)
                                                                                                                                  ∂x ∂y     ∂z
The continuity equation becomes                                                                                           ∂u    ∂u     ∂u ∂u              ∂p
                                                                                                                       ρ     +u    +v    +        = ρgx −    + µ∆u,                   (I 17)
                                          ∂ρ   3
                                                  ∂                                                                       ∂t    ∂x     ∂y ∂z              ∂x
                                             +       (ρui ) = 0
                                          ∂t i=1 ∂xi                                                                     ∂v    ∂v    ∂v    ∂v             ∂p
                                                                                                                     ρ      +u    +v    +w        = ρgy −    + µ∆v,                   (I 18)
                                      ∂
                                                                                                                         ∂t    ∂x    ∂y    ∂z             ∂y
Because i is repeated in the term ∂xi (ρui ), we make the rule that 3 is deleted, but
                                                                    i=1
                                                                                                                         ∂u    ∂w     ∂w ∂w               ∂p
remember it should be there whenever we see i repeated in a term. Thus, in Cartesian                                 ρ      +u    +v     +        = ρgz −    + µ∆w,                   (I 19)
tensor notation, the continuity equation is                                                                              ∂t    ∂x     ∂y   ∂z             ∂z

                                            ∂ρ   ∂                                             where ∆ is the Laplacian operator,
                                               +   (ρui ) = 0,
                                            ∂t ∂xi
                                                                                                                                         ∂2   ∂2  ∂2
or equivalently,                                                                                                                   ∆=        + 2 + 2.
                                                                                                                                        ∂x 2  ∂y  ∂z
                                               Dρ    ∂ui
                                                  +ρ     =0                            (I 8)
                                               Dt    ∂xi                                       I.5   Compressible Navier-Stokes equations
                                                                                               The compressible Navier–Stokes equations are required to describe high speed subsonic and
   We can also rewrite the expression for the stress tensor [Π] as                             hypersonic gas flows. These flows may generate temperature variations, so that the temper-
                                                 ∂uk          ∂ui ∂uj                          ature dependence of the material properties needs to be taken into account, i.e. µ, λ, ρ vary
                             Πij = (−p + λ           )δij + µ     +            ,       (I 9)   with (x, y, z, t) in general. In addition, for gases we have λ = − 2 µ from kinetic theory.
                                                 ∂xk          ∂xj   ∂xi                                                                                           3
                                                                                                  The constitutive relations are then reduced to:
                                                           δij = 1 if i = j,                                                          2 ∂uk         ∂ui ∂uj
where δij is the Kronecker symbol defined as
                                                           δij = 0 if i = j.                                               Πij = − p + µ    δij + µ     +          .                  (I 20)
                                                                                                                                      3 ∂xk         ∂xj   ∂xi
and so to avoid confusion with i and j, we can write
                                                                                                  The governing equations for the compressible flow of a perfect gas expressed with (x, y, z)
                                                            ∂uk                                coordinates are:
                                                     ·v =       .                     (I 10)
                                                            ∂xk
                                                                                               Continuity:
I.4     Incompressible Navier-Stokes equations                                                                                     Dρ    ∂u ∂v ∂w
                                                                                                                                      +ρ   +   +            =0                        (I 21)
                                                                                                                                   Dt    ∂x ∂y   ∂z
The constitutive relations are reduced to (ckeck as an exercise):
                                                                                               Momentum:
                                                             ∂ui ∂uj
                                    Πij = −pδij + µ              +                    (I 11)                               Du          ∂p
                                                             ∂xj   ∂xi                                                 ρ      = ρgx −
                                                                                                                           Dt         ∂x
   The incompressible Navier-Stokes equations are:                                                                               ∂      4 ∂u 2     ∂v ∂w
                                                                                                                              +      µ      −         +                               (I 22)
                                                                                                                                ∂x      3 ∂x 3     ∂y    ∂z
                                         ·v = 0                                       (I 12)                                     ∂      ∂u ∂v         ∂     ∂u ∂w
                                                                                                                              +     µ      +        +     µ    +
                                        Dv                                                                                      ∂y      ∂y ∂x         ∂z    ∂z   ∂x
                                      ρ     = ρg −            p + µ∆v,                (I 13)
                                        Dt
                                                       3                                                                                     4
                        Dv          ∂p
                      ρ    = ρgy −
                        Dt          ∂y
                              ∂      4 ∂v 2 ∂u ∂w
                           +      µ       −       +                                      (I 23)
                             ∂y      3 ∂y 3 ∂x       ∂z
                              ∂       ∂v ∂u       ∂      ∂v ∂w
                           +      µ      +     +      µ      +
                             ∂x       ∂x ∂y       ∂z     ∂z     ∂y
                        Dw          ∂p
                      ρ    = ρgz −
                        Dt          ∂z
                              ∂      4 ∂w 2 ∂v ∂u
                           +      µ        −      +                                      (I 24)
                             ∂z      3 ∂z    3 ∂y ∂x
                              ∂       ∂w ∂u        ∂     ∂w ∂v
                           +      µ      +      +      µ      +
                             ∂x       ∂x    ∂z    ∂y      ∂y     ∂z

Energy:

                  Dh   Dp   1  ∂   ∂h   ∂    ∂h   ∂    ∂h
              ρ      =    +      µ    +    µ    +    µ
                  Dt   Dt P r ∂x   ∂x   ∂y   ∂y   ∂z   ∂z
                                       2                  2                2
                           ∂u ∂v      ∂v ∂w          ∂w ∂u
                     + µ      +   +µ     +       +µ     +                                (I 25)
                           ∂y ∂x      ∂z    ∂y       ∂x   ∂z
                       4     ∂u ∂v ∂u    ∂v ∂w ∂v       ∂w ∂u                   ∂w
                     +   µ      −     +      −       +     −
                       3     ∂x ∂y ∂x    ∂y    ∂z ∂y    ∂z   ∂x                 ∂z

State equation:
                                                    γ p
                                            h=                                           (I 26)
                                                  γ −1ρ

h denotes the enthalpy of the system, γ = cp /cv is the ratio of specific heats at constant
pressure and constant volume. This parameter is characteristic of compressible gas flows
(e.g. for air, γ = 1.4). The Prandtl number given by,
                                                  µcp
                                           Pr =       ,
                                                   k
where k is the thermal conductivity, is a constant for each gas (e.g. for air Pr=0.7).




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