AAsymmetric Vortex Sheet
Alexander Migdal a) Department of Physics, New York University726 Broadway, New York, NY 10003
We present a steady analytical solution of the incompressible Navier-Stokes equation for arbitrary viscosity in an arbitrarydimension d of space. It represents a d − z . This profile is related to the Hermite polynomials H µ ( z ) which are analyticallycontinued to the negative fractional index µ = − dd − . In d = d ≥ w of the viscous lawyer will shrink to zero as ν for arbitrary dimension d >
3. In d = I. INTRODUCTION
First, let us define the notations. We are using the Einsteinconvention of summation over repeated indexes. The Greekindexes α , β , . . . with run from 1 to 3 and correspond to physi-cal space R and the lower case Latin indexes a , b , . . . will takevalues 1 , δ αβ , δ ab in three andtwo dimensions as well as the antisymmetric tensors e αβγ , e ab normalized to e = e = ∂ t (cid:126) v = ν (cid:126) ∇ (cid:126) v − ( (cid:126) v · (cid:126) ∇ ) (cid:126) v − (cid:126) ∇ p ; (1a) (cid:126) ∇ · (cid:126) v =
0; (1b) (cid:126) ω = (cid:126) ∇ × (cid:126) v ; (1c) H = (cid:90) d r (cid:126) v (1d)The general identity, which follows from the steady Navier-Stokes equation if one multiplies both sides by (cid:126) v and integratesover some volume V in space: − ν (cid:90) V d r (cid:126) ω = (cid:90) V d r ∂ β (cid:32) v β (cid:18) p + v α (cid:19) + ν v α ( ∂ β v α − ∂ α v β ) (cid:33) (2)By the Stokes theorem, the right side reduces to the flow overthe boundary ∂ V of the integration region V . The left side isthe dissipation in this volume, so we find: E = ν (cid:90) V d r (cid:126) ω = − (cid:90) ∂ V d (cid:126) σ · (cid:32) (cid:126) v (cid:18) p + v α (cid:19) + ν (cid:126) ω × (cid:126) v (cid:33) (3) a) Electronic mail: [email protected]
This identity holds for an arbitrary volume. The left siderepresents the viscous dissipation inside V , while the right siderepresents the energy flow through the boundary ∂ V .This formula is equivalent to the conventional representationof dissipation through the trace of the square of the straintensor: S αβ = ( ∂ α v β + ∂ β v α ) (4)The difference between the trace of the square of the vorticityvector and trace of the square of the strain tensor reduces tothe total derivative terms, which we prefer to include in thedefinition of the energy flow through the boundary by meansof Stokes theorem.The Euler equation corresponds to setting ν = ,with the Gaussian profile of vorticity in the normal directionand a constant tangent discontinuity ∆ v y = b in the Euler limit. (cid:126) v = {− ax , bS h ( z ) , az } ; (5a) (cid:126) ω = {− bS (cid:48) h ( z ) , , } ; (5b) S (cid:48) h ( z ) = h √ π exp (cid:32) − z h (cid:33) ; (5c) S h ( z ) = erf (cid:18) zh √ (cid:19) ; (5d) a = − ν h ; (5e)Our solution is different, being asymmetric and growing atinfinity in one direction. It remains to be observed in real fluidsor in DNS. II. STEADY SOLUTION FOR THE PLANAR SURFACE
Let us consider an infinite plane at z = R into upper and lower half spaces. a r X i v : . [ phy s i c s . f l u - dyn ] J a n We assume the separation of variables and write the follow-ing ansatz for the solution of the Navier-Stokes equations v i = S ( z ) ∂ i Γ ( x , y ) + ∂ i Φ ( x , y ) ; i = , v z = V ( z ) ; (7)We have from the incompressibility V (cid:48) ( z ) + S ( z ) ∂ i Γ + ∂ i Φ =
0; (8)This equation has the solution with linear V ( z ) , linear Γ ( x , y ) and quadratic Φ ( x , y ) . V ( z ) = az ; (9) Φ = − ax i ; (10) Γ ( x , y ) = q i x i (11)We wrote a Mathematica ® code to check the Navier-Stokesequation in an arbitrary dimension of space. In three dimen-sions, the Navier-Stokes equation leads to the following hyper-geometric equation for S ( z ) ν S (cid:48)(cid:48) − azS (cid:48) + aS = S ( z ) of this equation v z = az ; (13) v i = q i S − ax i ; (14) p = − a z − a x i , (15)For the negative a there is a solution of this equation, de-caying in the upper half space (we used here Mathematica ® tosolve the equation and analyze the solution) : a = − ν w ; (16) S ( z ) = F (cid:18) zw (cid:19) ; (17) F d ( z ) = − ( d − ) e − z Γ (cid:16) d − (cid:17) H − dd − (cid:16) z √ (cid:17) √ π (18)where H µ ( z ) is the Hermite polynomial, analytically continuedto the negative index µ using hypergeometric functions H µ ( z ) = µ √ π Γ (( − µ ) / ) F ( − µ / , / , z ) − µ √ π z Γ ( − µ / ) F (( − µ ) / , / , z )) (19)The normalization of S ( z ) is absorbed in the 2D vector param-eter (cid:126) q . The parameter w is arbitrary here, as well as (cid:126) q .Here is the plot of this function (Fig.1). - - - - - - - - FIG. 1. The velocity profile in normal direction. - - FIG. 2. The vorticity profile in normal direction.
The resulting formula for vorticity (cid:126) ω ( x , y , z ) ω z =
0; (20) ω x = − q y w F (cid:48) (cid:18) zw (cid:19) ; (21) ω y = q x w F (cid:48) (cid:18) zw (cid:19) ; (22)(23)At z → + ∞ , F (cid:48) ( z ) is exponentially small: F (cid:48) ( z → ∞ ) → exp (cid:16) − z (cid:17) √ z . (24)For negative z it decays as a square root, up to exponentialcorrection F (cid:48) ( z → − ∞ ) → √− z (25)Here is the plot of F (cid:48) ( z ) (Fig.2). It is asymmetric, with themaximum at z = − . III. DISSIPATION AND THE TURBULENT LIMIT
The dissipation integral logarithmically diverges E = ν (cid:90) V d r (cid:126) ω = ν Aq w (cid:0) log ( L / w ) + . + O ( / L ) (cid:1) (26)where A is the area of the plane and L is the depth of the lowerhalf space.We observe that the dissipation per unit area will be finiteprovided w log ( L / w ) ∝ ν (cid:126) q ; (27)On the other hand, we have another relation a = − ν w (28)These two relations describe different parameters of thetransverse velocity field inside the vortex sheet z ∼ wv i → q i F (cid:18) zw (cid:19) − ax i (29)For both terms in the velocity to scale the same way in thevortex sheet at vanishing viscosity, we need to have | (cid:126) q | ∝ | a | = ν w (30)Finally, plugging this formula into (27), we find the followingscaling relations between various parameters: w log ( L / w ) ∝ ν w ; (31) ν ∝ w (cid:0) log ( L / w ) (cid:1) − ; (32) w ∝ ν (cid:0) log ( L / w ) (cid:1) ; (33) | (cid:126) v | ∝ | a | ∝ | (cid:126) q | ∝ ν w − ∝ w − (cid:0) log ( L / w ) (cid:1) − ; (34) | (cid:126) ω | ∝ | (cid:126) q | w − ∝ w − (cid:0) log ( L / w ) (cid:1) − ; (35)Re ∝ a ν ∝ w − ∝ ν − (cid:0) log ( L / w ) (cid:1) − (36)These scaling laws are different from the ones suggested inmy previous work , where I was assuming an ordinary vortexsheet with Gaussian profile. The difference is only in thelogarithmic factors, the scaling exponents are the same.The outcome in the turbulent limit is qualitatively the same:the velocity scale grows, the width of the vortex sheet shrinks,and the Reynolds number Re grows in the turbulent limit.There is, however, one striking difference. There is no needfor explicit random force. The energy pumped equals to theenergy dissipated in any steady solution of the Navier-Stokesequations, including ours.Explicit formula (3) for the energy balance in this steadysolution shows that all dissipated energy is pumped in from theboundaries (the walls of the large cube where we are consid-ering our solution). Periodic boundary conditions are clearlyimpossible here, as there is a significant difference betweenfields in the upper and lower half-spaces. IV. SOLUTION FOR AN ARBITRARY DIMENSION
This solution directly generalizes to an arbitrary space di-mension d , with z being the normal direction to the d − x i , i = . . . d −
1. The solution of steady Navier-Stokes equations in d dimen-sions reads v z = az ; (37) v i = q i S − ax i d − p = − a z − a x i ( d − ) (39)with S ( z ) satisfying an equation ν S (cid:48)(cid:48) − azS (cid:48) + ad − S = S ( z ) = F d (cid:18) zw (cid:19) ; (41)(42)It is worth noting that the Fourier transform ˜ F d ( k ) of thisfunction F d ( z ) is an elementary function,satisfying the equation˜ F (cid:48) d ( k ) + (cid:18) k + d ( d − ) k (cid:19) ˜ F d ( k ) =
0; (43)˜ F d ( k ) = Ck − dd − exp (cid:32) − k (cid:33) ; (44)Our solution with exponential decay at positive z corre-sponds to a particular choice of the branch of the multival-ued function k − dd − , leading to the Hermite polynomial withnegative index (19).Note that the Fourier components of velocity and vorticityare heavily peaked around zero with effective width ∼ / w .This Gaussian tail is widening in the turbulent limit, beingreplaced by a power law.The singularity of the Fourier transform at k = d >
1, which reflects the growth of velocityat large coordinates. For the same reason, the total energy ofthis flow in a large box grows faster than volume.This is not a real obstacle, although, as the K41 scaling lawalso leads to growing velocity and the energy growing fasterthan volume. The difference is that here we have a velocityindex instead of the for K41.In two dimensions, this solution degenerates to F ( z ) = z .In this case, there is a constant vorticity and linear velocity inthe whole plane. This is a good reminder that 2D turbulence isvery different from the one we have in 3D.At d > z asexp (cid:16) − z (cid:17) . At large negative z it grows as F d ( z → − ∞ ) → − ( d − )( − z ) d − (45)The corresponding antisymmetric vorticity tensor ω αβ = ∂ α v β − ∂ β v α ; (46)(47)is vanishing unless when one of its two indexes equals d − ω ( d − ) i = − ω i ( d − ) = q i S (cid:48) ( z ) = q i w F (cid:48) d (cid:18) zw (cid:19) . (48)It does not depend upon the coordinates x i on the hyperplane,just as in the 3D case.Differentiating the (45) for S ( z ) we find for vorticity ω ( d − ) i ( z → − ∞ ) → q i w ( − z ) − dd − (49)This is a generalization of the square root decay we have in3D. Remarkably, d = d > z E = ν V d − q i w (cid:90) ∞ − ∞ d ξ ( F (cid:48) d ( ξ )) = const ν V d − q i w . (50)where V d − is the volume of the hyperplane.As a consequence, at d > V. CONCLUSION
We have found a steady analytic solution of the Navier-Stokes equation in an arbitrary dimension d of space. It repre-sents a d − | z | − dd − .In two dimensions, the solution degenerates to a constantvorticity. In higher dimensions, the vorticity is confined to the narrowlayer, with the width shrinking as a power of viscosity inthe turbulent limit. The d = z .The most remarkable thing is that it is an exact steady solu-tion of the Navier-Stokes equation for arbitrary viscosity andarbitrary space dimension. One could go from large viscosityto the turbulent limit of zero viscosity for a fixed energy flowand study the evolution of the flow along the way.At this time, such asymmetric vortex sheets have not beenobserved yet, neither in real fluids nor in DNS. The ones ob-served are better explained by the symmetric Burgers solution. ACKNOWLEDGMENTS
I am grateful to Dmytro Bandak for useful discussions andcomments.This work is supported by a Simons Foundation award ID686282 at NYU. M. Z. Bazant and H. K. Moffatt. Exact solutions of the Navier Stokesequations having steady vortex structures.
Journal of Fluid Mechanics , 541:55–64, October 2005. doi:10.1017/S0022112005006130. Alexander Migdal. Clebsch confinement and instantons in turbulence.
International Journal of Modern Physics A , 35(31):2030018, November2020. doi:10.1142/s0217751x20300185. URL https://doi.org/10.1142/s0217751x20300185https://doi.org/10.1142/s0217751x20300185