2.25 Advanced Fluid Mechanics, Solutions to the Final Exam

Solution to Problem 1-Final Exam 2rj g uj   


ρ, µ, σ ,u j h(r) u r1 r=R Figure 1: Viscous “Savart Sheet”. Image courtesy: Villermaux et. al. [1]. This kind of geometry was first studied by Felix Savart in 1833. (a): We know that the following independent properties are important for determining the value of R: radius of the disk (r1 ), radius of the jet (rj ), viscosity (μ), density (ρ), surface tension (σ), velocity of the jet (uj ), and gravity (g), so: R = f (r1 , rj , μ, ρ, σ, uj , g) thus, n = 8 and knowing that there are three dimensions involved in these parameters ([M ],[L], and [T ] ⇒ r = 3) we can conclude that there are 5 dimensionless groups. We select r1 ,vj , and σ as the repeating parameters and using Buckingham-Pi theorem following dimensionless groups will be identified: Π1 = R/r1 Π2 = rj /r1 Π3 = μuj /σ Π4 = gr1 /u2j Π5 = ρu2j r1 /σ So the relationship for R can be written in the following dimensionless form:   rj μuj gr1 ρu2j r1 R/r1 = f , , , σ r1 σ u2j (1) One can easily see that Π3 is the Capillary number (comparing the jet velocity with visco­ capillary velocity scale) and Π4 is the inverse of Froude number (comparing the gravity and inertia). The other dimensionless number found in this problem, (Π5 ) is indeed the Weber 1 number comparing the inertia stresses with capillary pressure. Thus, in order to ignore the effects of gravity and viscosity we have to satisfy the following1: gr1 << 1 u2j Ca for ignoring viscosity: Re ≡ << 1 We for ignoring gravity: F r−1 ≡ 2 Satisfying the two mentioned criteria, Equation 1 will be simplified to: ! rj ρu2j r1 R/r1 = f , σ r1 (2) (3) (b): From conservation of mass written for the dashed control volume in Figure 2 we will have (note that since viscous losses are ignored in this part, by Bernoulli, we can say that velocity at the edge of the disk is equal to the jet velocity i.e. along a streamline the velocity is unchanged.): Mass. Cons.: ρu2j (πrj2 ) = ρuj (2πr1 h(r1 )) → h(r1 ) = rj2 /2r1 (4) 2rj g  

proper s uj ρ, µ, σ ,u j h(r) u r1 r=R Figure 2: Simplified geometry with the selected control volume in dashed green line. For calculating the thrust force on the disc we need to apply the conservation of linear momen­ tum for y-component, lets assume that the disc is exerting F ey to the control volume: Momentum. Cons.: F ey = ρvj2 (πrj2 )ey 1 Another approach to do the dimensional analysis is picking ρ, σ, r1 as the repeating parameters and that √ will lead to R/r1 = g(rj /r1 , Oh = μ/ ρσr1 , Bo = ρgr12 /σ, W e = ρvj2 r1 /σ) and the criteria for ignoring gravity −1 and viscosity will respectively be F r = Bo/W e << 1 and Oh << 1; the physical reason that F r−1 << 1 means that gravity is negligible is the fact that inertia is fighting against gravity to keep the sheet flat rather than capillarity so the F r−1 << 1 is a physically more meaningful criteria for ignoring gravity effects compared to Bo << 1. 2 Stating the argument for neglecting the viscous effects by picking the Oh number or the Ca number is considered as the right answer but the most physical one is the Re << 1 2 thus an equal and opposite thrust force will be exerted from the liquid to the disc, i.e. FT ey = −ρvj2 (πrj2 )ey . (c): For finding a relationship for the value of R we have to consider a slice of the rim as our control volume (Figure 3) . The x-component of linear momentum enters the control volume at 1 but vanishes as the drops have zero/negligible velocity in the x-direction. This must happen from the force acting on the fluid in the control volume thus the conservation of linear momentum for the selected C.V. will be: W ρh(R)U 2 = 2W σ substituting U = uj and h(R) = rj2 /2R from conservation of mass: ρu2j rj2 /(2R) ρu2j rj R = 2σ ⇒ = 4σ rj (5) Rearranging the result in Equation 5 we can get the following form: ρu2j r1 R = 4σ r1 rj r1 2 which matches well with our predictions from dimensional analysis (equation 3). 

Fσ = 2π Rσ     
 


2   
ρ 2π Rh(R)U Fσ = 2π Rσ r~R Figure 3: A slice of the sheet rim picked as our control volume (with h << R) and length taken to be W = Rdθ. (d): Using the boundary layer scaling for diffusion of momentum away from the plate we have the following: J √ δ ∼ νt ∼ νr1 /uj and using the fact that h1 (inviscid) = rj2 /2r1 we will have: δ β≡ = h1 (inviscid) 4r13 ν uj rj4 ! 1/2 ⇒ β= 3 2 Rej r1 rj 3/2 (6) If we look into β through dimensional analysis then we will have: β=f ! ρu2j rj μ , Oh ≡ √ ,We ≡ ρσr1 (σ/r1 ) r1 and by rearranging the equation in (6) we will have: Oh1/2 β= W e1/4   3/2 r1 rj If viscous effects are important then the momentum of the incoming jet will get dissipated in the boundary layer and thus the liquid sheet emerging from the jet will be slower and thicker. (e): At the end of the disc there will be loss of momentum compared to the initial flux from the impinging jet due to the losses in the boundary layer. Through the concept of momentum thickness we know that the loss in the radial momentum at the edge of the disk should be: ΔM OM = Loss in the momentum = 2ρπr1 θu2j in which θ is the momentum layer thickness and for simplicity we can assume that θ and δ ∗ differ at most by a constant of order unity. After the sheet leaves the edge of the disc the velocity in the sheet becomes uniform and conservation of mass and momentum for a C.V. from the impingement point to r1 < r will give us two equations respectively: ρπrj2 u2j − 2ρπr1 δ ∗ uj2 = 2ρπrhU 2 ρπrj2 uj = 2ρπrhU by multiplying the momentum equation by (1/ρπrj2 uj = 1/2ρπrhU ) we will find an expression for U and then plugging that into the mass conservation equation we can find another expression for h. We rewrite the mentioned relationships in terms of β = δ/h1 = δ ∗ /(rj2 /r1 ): U = uj (1 − β) h = h(r)inviscid /(1 − β) = (rj2 /2r)(1 − β) (7) where h(r)inviscid = rj2 /2r is the variation in the sheet thickness observed in the inviscid case. U h(r) δ (r) r = r1 r Figure 4: Boundary layer developed on the disc. Since β > 0 then we can see that indeed the velocity of the sheet will decrease and the height will increase. 4 (f): Using the appendix in either Kundu or Panton we have:   ur 1 ∂uθ + τθθ = 2μ r ∂θ r knowing that by axisymmetry the first term on the right hand side is zero (there is no variation in the θ direction): U τθθ = 2μ r now we can compare the scale of inertial terms with the viscous terms in the liquid sheet: Ur inertia (1/2)ρU 2 ∼ = viscous 2μU/r 4ν if this ratio is larger than unity then viscous terms in the sheet are dominated by inertia terms. (g): We can repeat the same argument we had in part (c) but only this time we need to use new expression for U and h using equation (7): ρW U 2 (1 − β)2 hinviscid /(1 − β) = 2σW ⇒ R/rj = ρu2j rj (1 − β)/4σ (8) Which rearranging it to the dimensionless form will give: ρu2j r1 R = 4σ r1  rj r1 2 (1 − β) (9) using the expression for β we can rewrite the result: ρu2j r1  rj  2 R = 4σ r1 r1 Oh1/2 1− W e1/4  r1 rj  3/2 ! (10) which is exactly in accordance to what we expected from dimensional analysis in the absence of gravitational effects (part a). (h): By putting λcrit = R and using the result in equation (8) one can get: R= ρu2j rj2 (1 − β) ρ2 u4j rj2 (1 − β)3 10πσ 40π 2 10πσ → = → = 4σ σ2 ρa /ρ ρa uj2 (1 − β)2 ρa u2j (1 − β)2 which can be rewritten as: W ecrit (1 − β)3/2 = p 40π/α (11) in which α ≡ ρa /ρ and W ecrit ≡ ρu2j rj /σ. In the inviscid case for water in the air (ρa /ρ = 1.2/1000) we get W ecrit ∼ 900. As the effects of viscosity get larger and larger (β = 0) the critical Weber number shifts to higher values (Equation 11) and the rim radius at a given value of the Weber number is reduced, from equation (8). Interestingly because the dependence of equation (11) on β is stronger than equation (8) this means the viscous Savart sheet actually extends a little bit further out radially before it starts flapping (Figure 5). 5 

  
 

Figure 5: Copy of Figure 4 from Villermaux’s paper ([1]). References [1] E. Villermaux, V. Pistre, and H. Lhuissier. The viscous Savart sheet. Journal of Fluid Mechanics, 730:607–625, August 2013. 6 2.25 – Fluid Mechanics Final Exam, problem 2 solution Part (a) (1 point) 2 L Dimensions of w :   T  L Dimensions of U  :   T  Dimensions of a 2 and z 2:  L2   L2   L         L2    T  T   2  1    1 2 Part (b) (1 point) 2 1.5  1  y/a 0.5 0 -0.5 -1 -1.5 -2 -4 -3 -2 -1 0 1 2 3 4 x/a Figure 1: A holiday-themed plot of velocity potential contours (green) and streamlines (red) for potential flow around a flat plate of length 2a and negligible thickness. Lines of constant potential and streamlines are everywhere perpendicular, except at the stagnation point at the origin. Flow is from left to right. Note: the vertical “streamlines” at x = 0 are not physical; they are artifacts of discontinuities in the stream function at x = 0. 7 Part (c) (3 points) The complex velocity is U   x  iy  U  z dw  vx  ivy   dz z 2  a2 2 x  2ixy  y 2  a 2 . (1.1) To find vx along the centerline, set y = 0 and isolate the real part of dw / dz: vx  y  0   U  x x2  a2 . (1.2) Since vx is positive everywhere (in the reference frame of the plate),   vx  y  0      U x x2  a2 U  x x2  a2 , x0 (1.3) , x0 Similarly, the y-direction slip velocity on the plate is determined by setting x = 0 and looking at the imaginary part of dw / dz: vy  x  0  U y a2  y2 . (1.4) Again, use physical intuition to determine the sign of vy:  U y ,  2 2  a y v y  x  0, a  y  a     U  y ,  a2  y2  x  0 (left side of plate) (1.5)  x  0 (right side of plate) Figure 2 shows the signs of the velocity components in each of the four quadrants. II vx > 0, vy > 0 I vx > 0, vy < 0 III vx > 0, vy < 0 IV vx > 0, vy > 0 Figure 2: Signs of velocity components in each quadrant. 8 Part (d) (2 points) There are at least two curves that may be used to compute the circulation around the top half of the plate. Curve 1 The simplest curve to use is likely the rectangle ABCD shown below. Figure 3: Counter-clockwise curve used for computing circulation. The circulation is determined from its definition   ABCD v  dr   v  dr   v  dr   v  dr   v  dr AB BC CD (1.6) DA 0 0 where v is the velocity vector and dr is a differential line segment along ABCD. The integrals along BC and DA vanish because the plate is infinitely thin. The integrals along AB and CD reduce to  a U  ydy 0 a2  y2  v  dr   v  dr   AB CD    2U   a 0  ydy a2  y2 a 0 U  y  dy  a2  y2 (1.7) . This integral may be evaluated by, for example, making the substitution u  a 2  y 2 , du  2 ydy to obtain   2U  a. 9 (1.8) Curve II One can also integrate along a rectangle that includes the entire half-domain y > 0. This rectangle extends to infinity in the +x, +y, and –x directions and coincides with the x-axis. Since the fluid contains no vorticity flux (it is a potential flow, which by definition is irrotational), all of the vorticity within curve II is contained in the top half of the plate. Thus, the answer for circulation should be the same as for curve I. Figure 4: The desired circulation may also be computed by integrating around the entire top half of the domain. The result for circulation is the same because the flow around the plate is irrotational, and so the entire vorticity flux is “bound into” the top half of the plate. In this case, the y-components of velocity along the left and right segments of the curve are zero (since v  Ue x as distance from the plate approaches infinity). The circulation becomes        U  dx   U     x dx  U     1 dx 2 2  x a  x a  x 2 2 (1.9) By symmetry, this integral can be written L   x   2U     1 dx  2U  lim  x 2  a 2  x   2U  a. 2 2 0 L   0  x a  (1.10) Since the flow is symmetric about the x-axis, the circulation around the bottom half of the plate is equal and opposite to (1.8). Thus the net circulation around the entire plate is zero. 10 Part (e) (2 points) It was given that the flow is incompressible (and thus barotropic), inviscid, and free of body forces. If we ignore any flows induced by the removal of the oar, we can thus apply Kelvin’s Circulation Theorem and argue that the circulation around the top and bottom halves of the oar is conserved, and the strength of both resulting vortices is given by the circulation found in part (d): bottom  top  2U  a. (1.11) The vorticity may be computed from Stokes’ Theorem. Consider the bottom vortex, which has positive circulation: bottom  2U  a   z dA  z R 2  z ,bottom  z ,top  A 2U  a .  R2 (1.12) Part (f) (2 points) Since the vorticity is uniformly distributed in each vortex, the fluid within the vortices is moving in solid-body rotation. The velocity field inside the bottom vortex (ignoring flows induced by the top vortex) is given by v  z 2 r Ua r.  R2 (1.13) Ua . R (1.14) Evaluating at r = R, v  r  R   One can also arrive at this answer by assuming the flow field outside the vortex to be that of an irrotational vortex  v   / 2 r  , setting r = R, and using the result from part (e): v  r  R   2U a U a      . 2 R 2 R  R (1.15) The pressure is obtained from applying the Bernoulli equation between a point at infinity (where the fluid is stationary) and the edge of the vortex. Recall that the Bernoulli equation may be applied between any two points since the flow outside the vortices is irrotational. Since there are no body forces to consider, the Bernoulli equation reads 11 p  1 2   0 2 fluid is stagnant at infinity in the laboratory reference frame 1  p  r  R    v2  r  R  2 (1.16) Note that we are considering each vortex to be isolated from the other, i.e. the velocity field induced by the other vortex is ignored in formulating (1.16); we will justify this assumption later. Solving for the desired pressure, 2 1 U a  p  r  R   p      . 2  R  (1.17) Part (g) (1 point) In this part we consider each vortex as an irrotational vortex (i.e., one that can be described by a velocity potential), but now we specifically account for the influence of the other vortex. The velocity field associated with an irrotational vortex is v   U  a  , r  R. 2 r r (1.18) Each vortex will induce the other one to move. Each vortex core moves at a speed given by evaluating (1.18) at r = H: v  r  H   U  a . H (1.19) In the laboratory reference frame, the vortices propagate to the left. Thus, V Ua from right to left. H (1.20) Equation (1.20) justifies the assumption made in part (f) that the velocity field induced by the other vortex may be neglected when computing the tangential velocity at the edge of the vortex (r = R). From part (f), the tangential velocity at the edge of the vortex scales as 1/R. Equation (1.20) shows that the contribution to the tangential velocity due to the other vortex will scale as 1/H. Since H R, it is therefore safe to assume that V 12 v  r  R  . Part (h) (4 points) The unsteady viscous decay of a free vortex was covered in detail on problem set 8. The governing equation is the Navier-Stokes equation in the -direction, which in our case reduces to v  1     rv  . t r  r r  (1.21) The necessary assumptions are that the flow is:  Axisymmetric    /   0   Unaffected by the velocity field due to the other vortex  Unidirectional  vr  vz  0  Initial condition: Initially, the flow field is that of an irrotational vortex: v  r , t  0    . 2 r (1.22) Here   2U  a is the circulation around the bottom vortex found in part (e). Boundary conditions (valid for t > 0): (1) The velocity is zero in the center of the vortex (similar to solid-body rotation), and (2) very far away the velocity field looks like an irrotational vortex (because the fluid very far away from the vortex core has not yet felt the influence of viscosity): v  r  0, t   0, t  0 v  r  , t    , t 0 2 r (1.23) Following the problem statement, introduce the dimensionless velocity and similarity variable v r2 F ,  .  / 2 r 4 t (1.24) Using the chain rule, we can transform the partial derivatives with respect to t and r in (1.21) to ordinary derivatives with respect to :     d  r 2 d    , t t d 4 t 2 d     d  r d     . r r d 2 t d 13 The left and right-hand sides of equation (1.21) are then rewritten as v  r 2 dF  , t 2 r 4 t d   1     1 dF   r d 2F    rv   .     r  r r r  2 2 t d  4 t 2 t d 2  The governing equation (1.21) then becomes r dF r d 2 F  8 t d 8 t d 2  d 2 F dF   0. d 2 d (1.25) (1.26) The boundary and initial conditions collapse to F   0   0, F      1. (1.27) The solution can be found by several different methods. Here, we write the characteristic equation for the ODE (1.26) as s 2  s  s  s  1  0  s  0, 1. Then the general solution is F    A  Be . From (1.27), we have A  B  1 and thus F    1  e , (1.28)  r 2  Ua  v  r , t   1  exp    . r   4 t   (1.29) or, in dimensional terms, This result implies that the propagation speed V of the vortices decays with time, according to V t    H 2  Ua  1  exp    . H   4 t   14 (1.30) Part (i) (1 point) From the solution to part (h), it is straightforward to determine the vorticity distribution as a function of space and time: z  r , t    r2  U a  r2  U a 1  1 v  rv   r   exp      exp   2  , r r r  2 t  4 t  2 t    0 (1.31) The vorticity has a normal distribution centered at the vortex core with standard deviation . Note that the peak value (at r = 0) changes with time. It is convenient to define the length scale   2 t. Even without knowing the solution to (h), one can still make an argument based on previous problems we have considered throughout the course that deal with viscous diffusion of vorticity (e.g., Rayleigh problem, boundary layers, etc.). From a similar argument one could conclude that  ~  t . Part (j) (3 points) The vorticity distributions will begin to overlap when their characteristic widths are approximately equal to H/2. This happens at a characteristic time tc, where H H2   tc  tc ~ . 2 4 (1.32) The vortex propagation speed scales with the value found in part (g): V~ Ua . H (1.33) Then the distance x that the vortices travel before they decay away is roughly the product of V and tc: x  Vtc  U  a H 2  U  H  a   H 4     4  ReH x    a ~ ReH .  (1.34) A Reynolds number clearly emerges from (1.34). This is not surprising, as the Reynolds number should govern the distance that the vortices propagate before they are spread out by viscosity. Thus, in this case the Reynolds number can be interpreted as a dimensionless propagation distance. 15