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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). 5