3 The boundary layer equations

Transcription

1 3 The boundar laer equations Having introduced the concept of the boundar laer (BL), we now turn to the task of deriving the equations that govern the flow inside it. We focus throughout on the case of a 2D, incompressible, stead state of constant viscosit. We start for simplicit b considering a flat plate of length L with normal in the direction, subject to a uniform free stream far from the plate of velocit U in the x direction. We return below to discuss the extension of our results to curved surfaces. U U x Figure 5: The boundar laer along a flat plate at zero incidence. (After Schlichting, Boundar Laer Theor, McGraw Hill.) 3.1 Derivation = rescale 1 L x 1 x = x L Figure 6: Boundar laer transformation, to render the flow variables O(1). Dot dashed line shows the thickness of the boundar laer. Looking back at the derivation of the non-dimensional N S Eqns. 26 to 28, we recall that we expressed all lengths and velocities in the natural units L and U. The ke point in boundar laer theor, however, is the existence of two different length scales: the characteristic size of the object L, the tpical thickness of the boundar laer L. Recall Fig. 4. (The smallness of will emerge from the analsis that follows.) Accordingl, we now take L as the natural unit of length in the x direction, and as that in the direction, 11

2 defining: x = x L, =. (36) The actual thickness of the BL will actuall var with distance along the plate (Figs. 5 and 6; not shown in Fig. 4), so we now choose to be the value at the trailing edge, x = L, Fig. 6. Thus, both L and are constants in what follows. Similarl, we expect different scales for the x and components of velocit, and choose: u = u U, v = v V, (37) where U and V are likewise constants. and V are as et unknown: the will be determined b the following analsis. Finall we adimensionalise the pressure in the usual wa: P = P ρu2. (38) The basic strateg in such a transformation is to render all flow variables x,, u, v, P and all derivatives ( u / x, etc.) of order unit, Fig. 6. B expressing the flow equations in terms of these rescaled variables, an terms that are negligible will then reveal themselves b having a small prefactor. (This was precisel the procedure adopted to identif regimes of high and low Re in Sec. 2.1.) We this in mind, we now appl the transformation 36 to 38 to each of the flow Eqns. 22 to 24 in turn. Continuit: In terms of the above variables the continuit equation, Eqn. 22, becomes U u L x + V For u / x and v / to both be of order unit, we require v = 0. (39) L V U = O(1), and so choose V = U L. (40) The non-dimensional continuit equation is then u x + v = 0. (41) Anticipating the result (derived below) that the BL thickness is small, we see from Eqn. 40 that the characteristic vertical velocit V is also small, as we would expect intuitivel. x-momentum The horizontal component of the force balance equation, Eqn. 23, becomes ρu 1 L Uu u x + ρu 1 L Uv u = ρ U2 P ( L x + µ U 1 2 u L 2 x 2 + U 1 2 u ) 2 2. (42) On dividing through b ρu 2 /L, we obtain the non-dimensional form u u x + v u = P x + 1 Re 12 2 u (L/)2 + x 2 Re 2 u, (43) 2

3 in which the Renolds number Re = ULρ/µ, as usual. In the limit Re, the coefficient of 2 u / x 2 becomes small, so this term can be ignored. In order for the remaining viscous term to be retained in accordance with the discussion of Sec. 2, the coefficient of 2 u / 2 must remain of order unit. Therefore we require L = O(Re 1/2 ), and so choose L = Re 1/2. (44) The x momentum equation then becomes u u x + v u = P x + 2 u. (45) 2 Eqn. 44 is an important result: it tells us that the BL becomes infinitesimall thin in the limit Re, as anticipated above. Euler s equations of inviscid flow, valid outside the BL, therefore appl right down to (an infinitesimal distance above) the bod surface. This will allow us below to find an expression for the term P / x on the RHS of Eqn momentum: The vertical component of the force balance equation, Eqn. 24, becomes ρu 1 L U v u L x + ρu L 1 U L v v = ρ U2 P + µ On division throughout b ρu 2 /, we obtain ( U L 1 2 v U L 2 + x 2 L 1 2 v ) 2 2. (46) ( 2 } {u L) v x + v v = P + 1 { ( ) 2 2 v Re L x v } 2. (47) Neglecting small terms, O(Re 1 ), we get simpl 0 = P. (48) This tells us that the pressure does not var verticall through the BL: P = P (x ) onl. The BL therefore experiences, through its entire depth, the pressure field that is imposed at its exterior edge, P (x ) = P e(x ). On the exterior, the flow is governed b Euler s inviscid equations. As noted above, we solve these in a domain that continues essentiall right down to the solid surface, since the BL is infinitesimall thin. The BL however plas the crucial role of rendering Euler s equations free of the no-slip condition, Fig. 4 (leaving onl no-permeation). For our purposes, we assume that this inviscid calculation has alread been performed, giving u = u e(x ),v = 0 on the exterior edge of the BL: i.e., the function u e prescribing the inviscid slipping velocit is given to us. P (x ) = P e (x ) can be expressed in terms of u e via the x component of the inviscid momentum equation, Eqn. 34 u du e e dx = dp e dx. (49) 13

4 Substituting this into Eqn. 45, and recalling Eqn. 41, we get finall the adimensional boundar laer equations: u x + v = 0, (50) u u x + v u with the transformation variables and boundar conditions: x = x L, = Re 1/2 L, u = u U, v = Re 1/2 v U = u du e e dx + 2 u, (51) 2 Re = UL ν On the bod surface, = 0: no slip and no permeation, u = v = 0. At the exterior edge of the boundar laer 3,, the velocit must match the surface slipping velocit calculated according to inviscid theor: u u e(x ). Because the boundar laer equations are independent of Re, the onl information required to solve them is u e (x ), which depends on the shape of the bod and its orientation relative to the free stream The stream function The equations just derived can be simplified b noticing that the continuit equation is automaticall satisfied (in 2D) if we define a stream function Ψ such that (52) u = Ψ, v = Ψ x. (53) Plug these into Eqn. 22 to verif this. In dimensionless form in which Ψ is defined b u = Ψ, v = Ψ x, (54) Ψ = Ψ Re1/2 UL = Ψ νul. (55) (Check this scaling of Ψ b plugging the usual transformations x = x L, = L/Re 1/2, u = u U, v = v U/Re 1/2 together with Ψ = Ψ /α into Eqn. 53 to see that we get Eqn. 54 provided α matches the prefactor in Eqn. 55.) Having (automaticall) solved the continuit equation 50, we now just need to solve the momentum equation 51. Expressed in terms of the stream function, this is: The boundar conditions are, as usual: Ψ 2 Ψ x Ψ 2 Ψ x 2 = du u e e dx + 3 Ψ. (56) 3 3 The limit, which takes us to the inviscid side of the infinitesimall thin boundar laer, is ver different from the limit, which trul takes us far from the bod, out to the free stream. 4 Strictl, the equations just derived become exact onl in the limit Re. For large Re, the represent a first order approximation. The scaling used to obtain them implies an O(Re 1/2 ) correction in the inviscid region, for example. 14

5 No slip and no permeation at the solid surface, Ψ = Ψ x = 0 at = 0. (57) Inviscid slipping solution on the exterior edge of the boundar laer, u = Ψ u e(x ) as. (58) We will emplo the stream function extensivel throughout the course. 3.3 Curved surfaces The boundar laer equations just obtained appear rather restrictive, because the were derived for a flat surface. What about curved surfaces? Provided remains small compared with the local radius of curvature of the surface, it is possible to show that Eqns. 50 and 51 still appl. The onl modification is geometrical: x becomes the coordinate measured along the curved surface and the normal to the local surface direction. See figure 7 (which also shows the wa in which the boundar laer can continue as a wake behind the obstacle). U x aerofoil boundar laer and wake Figure 7: Boundar laer around a curved surface (and the wake behind it). 3.4 Forces on the bod We now consider the forces exerted b the flow on the bod. These are calculated b integrating the normal and tangential stress components over the surface to find the lift and drag forces respectivel. We recall that the dimensional stress tensor, Eqn. 17 can be written as ( ui Π ij = P ij + p 0 ij + µ + u ) j. (59) x j x i When integrated over the surface, the static pressure p 0 ields a buoanc force ρgv b (where V b is the volume of the bod) that is generall insignificant, at high Re, compared with the other forces of lift and drag. In an case, it is independent of the flow, and we neglect it. The (remaining) stress tensor ma be non-dimensionalised using the boundar laer transformations 52 to give Π xx ρu 2 = P + 2 u Re x, 15 Π ρu 2 = P + 2 v Re, (60)

6 For large Re, these reduce to Π x ρu 2 = Π x ρu 2 = 1 ( u Re 1/2 + 1 Re v x ). (61) Π xx ρu 2 = Π ρu 2 = P, Π x ρu 2Re1/2 = u. (62) The normal stresses are thereb seen to reduce to the pressure P, and the viscous shear stress Π x to µ u/. The skin friction coefficient at the bod surface = 0 is defined as a measure of the shear stress relative to the natural pressure scale ρu 2 : c f = (Π x) =0 1 = µ( u/ ) =0 2 ρu2 1. (63) 2 ρu2 (The factor half is for convention.) Using transformations 52 we see ( u c f Re 1/2 ) = 2 =0 = f(x ) alone, for geometricall similar bodies. (64) 16