We have successfully attempted to solve the equations of full-MHD model within the framework of Ψ - ω formulation with an objective to evaluate the performance of a new higher order scheme to predict better values of control parameters of the flow. In particular for MHD flows, magnetic field and electrical conductivity are the control parameters. In this work, the results from our efficient high order accurate scheme are compared with the results of second order method and significant discrepancies are noted in separation length, drag coefficient and mean Nusselt number. The governing Navier-Stokes equation is fully nonlinear due to its coupling with Maxwell’s equations. The momentum equation has several highly nonlinear body-force terms due to full-MHD model in cylindrical polar system. Our high accuracy results predict that a relatively lower magnetic field is sufficient to achieve full suppression of boundary layer and this is a favorable result for practical applications. The present computational scheme predicts that a drag-coefficient minimum can be achieved when β =0.4 which is much lower when compared to the value β=1 as given by second order method. For a special value of β =0.65 , it is found that the heat transfer rate is independent of electrical conductivity of the fluid. From the numerical values of physical quantities, we establish that the order of accuracy of the computed numerical results is fourth order accurate by using the method of divided differences.
The calculation and prediction of physical quantities to the desired accuracy is necessary in every engineering design because otherwise an under-performance together with an overshoot in the design cost. The study of magnetohydrodynamic flows with its heat transfer is important due to many practical applications in science and engineering. For instance, in many semiconducting single crystal growth processes, especially using the Czochralski technique, cylindrical shaped crystal is grown from the melt containing the material in the molten (liquid) state. In such systems, calculation of heat transfers is fairly complex due to the nonlinear coupling of the momentum with the heat diffusion and advection apart from radiative transfers. These semiconducting crystals are mostly used as substrates in electronic industry. However, during the crystal growth process, thermal stresses will be present if heat transfer is not well controlled [
Many industrial applications include liquid metals in metallurgical processes such as stirring, pumping, casting, suppressing etc. [
The solution methodology used for MHD flows can be classified in to three categories namely a) getting approximate analytical solutions by perturbation techniques or otherwise, b) getting the boundary layer solution and c) obtaining the full solution in the entire domain of interest. In the case of boundary layer solution approach, the solutions are obtained within the boundary layer of the flow, and the detailed flow velocities and temperature will not be available in the domain of fluid flow. Under the first category a), limited attempts are made using Laplace transform technique [
In order to study the steady state flow properties and heat transfer in flows of electrically conducting fluids in presence of magnetic field, the governing equations would be the modified Navier-Stokes equation (in which additional body force terms due to magnetic field) which is coupled with the Maxwells equations of electrodynamics together with energy equation. The non- dimensionalized equations under consideration are
∇ ⋅ q = 0 (1)
( q ⋅ ∇ ) q = − ∇ p + 2 R e ∇ 2 q + β ( j × H ) (2)
∇ ⋅ H = 0 (3)
∇ × H = j + ∂ D ∂ t (4)
∇ ⋅ j = 0 (5)
j = R m 2 [ E + ( q × H ) ] (6)
( q ⋅ ∇ ) T = 2 R e P r ∇ 2 T (7)
where the following definitions are used in the process of non-dimensionalization.
q = q ′ U ∞ , p = p ' ρ U ∞ 2 , r = r ′ a , T = T ′ − T ∞ T s − T ∞ ,
H = H ′ H ∞ , E = E ′ E ∞ , j = j ′ j ∞
in which the primed variables denote the respective quantities with dimensions. The Alfvén number β is a dimensionless quantity characterizing the flow in presence of magnetic field and it is the ratio of the speed of the Alfvén wave to the speed of the main stream fluid. The last term in Equation (2) is the body force term which is nonlinearly coupled with other equations in the list. Equation (1) is due to incompressibility condition while Equations ((3) and (4)) are the Maxwell’s equations to be satisfied for the applied magnetic field H . In the Amperes law (4), the second term is due to electric displacement vector D . Since displacement current density is negligible in fluid flows the second term in the right hand side of Equation (4) can be dropped. For steady state conditions, the electrodynamic continuity equation is (5) and the conduction current density j has to satisfy the Ohms law which is Equation (6). First it is noted that Equations ((1) to (6)) are coupled. Now, if heat transfer analysis is to be carried out, then the energy transport equation to be solved is given by Equation (7). If we solve (7) by treating the velocity q as coupled with Equations ((1) to (6)) then we can get the natural convection heat transfer properties. On the other hand, if we first solve the set of Equations ((1) to (6)) and provide this solution to Equation (7) as its input, then we can study the forced convection properties. In any case, first we need a discretization scheme to solve the governing equations. In the following, we propose to device a solution method based on streamfunction-vorticity approach and which is suitable for two- dimensional flow simulations. In cylindrical polar system, the velocity and applied magnetic field are
q = ( q r , q θ ,0 ) (8)
H = ( h r , h θ ,0 ) (9)
To satisfy the incompressibility condition, the velocity components can be expressed in terms of a scalar streamfunction ψ given by
q r ( r , θ ) = 1 r ⋅ ∂ ψ ( r , θ ) ∂ θ (10)
q θ ( r , θ ) = − ∂ ψ ( r , θ ) ∂ r (11)
Since Equations ((1) and (3)) exhibit solenoidal nature, a definition similar to (10) and (11) can be used by making use a scalar called magnetic streamfunction, denoted by A as below.
h r ( r , θ ) = 1 r ⋅ ∂ A ( r , θ ) ∂ θ (12)
h θ ( r , θ ) = − ∂ A ( r , θ ) ∂ r (13)
By taking curl of Equation (8), the vorticity is obtained as
∇ × q = ω (14)
which, upon using (10) and (11), we get the differential equation for velocity as below.
q θ r + ∂ q θ ∂ r − 1 r ( ∂ q r ∂ θ ) = ω (15)
and where ω has components only in the z-direction and is a function of r and θ of the cylindrical polar coordinate system. We make use of Equation (14) in (2) by applying the curl operator to (2) which results in
− ∇ × ( q × ω ) = − 2 R e ⋅ ∇ × ( ∇ × ω ) + β ⋅ [ ∇ × { ∇ × H } × H ] (16)
where we have also used (4) after neglecting the displacement vector. In the present work, we consider the case of applying only an external magnetic field but no electric field is applied. In addition, for the case of two-dimensional flows, it can be shown that E = 0 if only magnetic field is applied to the electrically conducting flow. Then, from (4) and (6), we have
∇ × H = R m 2 ⋅ ( q × H ) (17)
Using (17) in (16) and then expanding the curl operator in the cylindrical system and by making use of (9) we obtain the differential equation for vorticity as given below.
[ q r ( ∂ ω ∂ r ) + q θ r ( ∂ ω ∂ θ ) ] − 2 R e [ ∂ 2 ω ∂ r 2 + 1 r ( ∂ ω ∂ r ) + 1 r 2 ( ∂ 2 ω ∂ θ ) ] = β R m 2 ⋅ [ − h r 2 q θ r − 2 h r q θ ( ∂ h r ∂ r ) − h r 2 ( ∂ q θ ∂ r ) + h r h θ q r r + h θ q r ( ∂ h r ∂ r ) + h r q r ( ∂ h θ ∂ r ) + h r h θ ( ∂ q r ∂ r ) + 2 h θ q r r ( ∂ h θ ∂ θ ) + h θ 2 r ( ∂ q r ∂ θ ) − h r h θ r ( ∂ q θ ∂ θ ) − h θ r ( ∂ h r ∂ θ ) − q θ h r r ( ∂ h θ ∂ θ ) ] (18)
Now, if we simplify the Equation (17) using (8) and (9) we get the differential equation for magnetic field as follows.
h θ r + ∂ h θ ∂ r − 1 r ⋅ ( ∂ h r ∂ θ ) = R m ( q r h θ − q θ h r ) 2 (19)
Similarly, Equation (7) is simplified using (10) and (11) to yield the differential equation for temperature as given below.
q r ( ∂ T ∂ r ) + q θ ( ∂ T ∂ θ ) = 2 R e ⋅ P r [ ∂ 2 T ∂ r 2 + 1 r ( ∂ T ∂ r ) + 1 r 2 ( ∂ 2 T ∂ θ 2 ) ] (20)
Finally we have a set of four coupled partial differential Equations ((15), (18)-(20)) having several nonlinear terms in the vorticity differential equation. The components of q and H appearing in these four equations are given by (10) to (13). The above set of equations are generally valid for MHD flows in cylindrical geometry.
In this section we outline the discretization procedure for the discretization of governing equations. The implementation of fourth order finite differences to the governing equations for the present problem requires considerable exercise because of curvilinear coordinate system. From the Taylor series, for the x and h-direction, if we denote h and k as the grid spacing along radial and angular direction then, the fourth order accurate finite differences for the first and second derivatives can be written as
∂ Ψ ∂ ξ = D ξ Ψ − h 2 6 ∂ 3 Ψ ∂ ξ 3 + O ( h 4 ) (21)
∂ 2 Ψ ∂ ξ 2 = D ξ 2 Ψ − h 2 12 ∂ 4 Ψ ∂ ξ 4 + O ( h 4 ) (22)
∂ Ψ ∂ η = D η Ψ − k 2 6 ∂ 3 Ψ ∂ η 3 + O ( k 4 ) (23)
∂ 2 Ψ ∂ η 2 = D η 2 Ψ − h 2 12 ∂ 4 Ψ ∂ η 4 + O ( k 4 ) (24)
where D ξ , D ξ 2 , D η and D η 2 denote the standard second order central difference operators which are given by
D ξ Ψ i , j = Ψ i + 1 , j − Ψ i − 1 , j 2 h (25)
D ξ 2 Ψ i , j = Ψ i + 1 , j − 2 Ψ i , j + Ψ i − 1 , j h 2 (26)
D η Ψ i , j = Ψ i , j + 1 − Ψ i , j − 1 2 k (27)
D η 2 Ψ i , j = Ψ i , j + 1 − 2 Ψ i , j + Ψ i , j − 1 k 2 (28)
The second order central difference operators for cross-derivatives can be obtained from Taylor series, and are given by
D ξ D η Ψ i , j = 1 4 h k [ Ψ i + 1, j + 1 − Ψ i + 1, j − 1 − Ψ i − 1, j + 1 + Ψ i − 1, j − 1 ] (29)
D ξ 2 D η Ψ i , j = 1 2 h 2 k [ Ψ i + 1 , j + 1 − Ψ i + 1 , j − 1 + Ψ i − 1 , j + 1 − Ψ i − 1 , j − 1 − 2 ( Ψ i , j + 1 − Ψ i , j − 1 ) ] (30)
D ξ D η 2 Ψ i , j = 1 2 h k 2 [ Ψ i + 1 , j + 1 + Ψ i + 1 , j − 1 − Ψ i − 1 , j + 1 − Ψ i − 1 , j − 1 − 2 ( Ψ i + 1 , j − Ψ i − 1 , j ) ] (31)
D ξ 2 D η 2 Ψ i , j = 1 h 2 k 2 [ Ψ i + 1, j + 1 + Ψ i + 1, j − 1 + Ψ i − 1, j + 1 + Ψ i − 1, j − 1 − 2 ( Ψ i + 1, j + Ψ i − 1, j + Ψ i , j + 1 + Ψ i , j − 1 ) + 4 Ψ i , j ] (32)
Equation (15) along with definitions (10) and (11) gives
1 r [ ∂ ψ ∂ r ] + ∂ ∂ r [ ∂ ψ ∂ r ] − 1 r [ ∂ ∂ θ { 1 r ∂ ψ ∂ θ } ] = ω (33)
Now the transformation r = e π ξ and θ = π η are implemented followed y the usage of D-operator from Equations ((21) to (28)) so that the above equation becomes.
D ξ 2 ψ i , j + D η 2 ψ i , j + s i , j = ( h 2 12 ∂ 4 ψ ∂ ξ 4 + k 2 12 ∂ 4 ψ ∂ η 4 ) i , j + O ( h 4 , k 4 ) (34)
where the RHS of the above equation is the truncation error term and O ( h 4 , k 4 ) indicates that the above equation is of the order of h 4 in ξ direction and of the order of k 4 in η direction. and
s i , j = π 2 e 2 π ξ ω i , j (35)
Since the Equation (34) inherently contain higher order derivatives of the stream function, they are computed as
∂ 3 ψ ∂ ξ 3 = − ∂ 3 ψ ∂ ξ ∂ η 2 − ∂ s ∂ ξ = − D ξ D η 2 ψ − D ξ s (36)
∂ 4 ψ ∂ ξ 4 = − ∂ 4 ψ ∂ ξ 2 ∂ η 2 − ∂ 2 s ∂ ξ 2 = − D ξ 2 D η 2 ψ − D ξ 2 s (37)
∂ 3 ψ ∂ η 3 = − ∂ 3 ψ ∂ ξ 2 ∂ η − ∂ s ∂ η = − D ξ 2 D η ψ − D η s (38)
∂ 4 ψ ∂ η 4 = − ∂ 4 ψ ∂ ξ 2 ∂ η 2 − ∂ 2 s ∂ η 2 = − D ξ 2 D η 2 ψ − D η 2 s (39)
Substituting Equations (36)-(39) in the discretized streamfunction Equation (34) and applying the D-operator from Equations (25)-(32), we get the fourth order accurate discretized representation of Equation (33) as given below.
2 ( 2 z − h 2 − k 2 ) ψ 0 + ( k 2 − 2 z ) ( ψ 1 + ψ 3 ) + ( h 2 − 2 z ) ( ψ 2 + ψ 4 ) + z ∑ i = 5 8 ψ i = − h 2 k 2 π 2 e 2 π ξ { ( 1 + 4 π 2 x ) ω 0 + 4 π x D ξ [ ω ] + ( x D ξ 2 + y D η 2 ) [ ω ] } (40)
where x = h 2 / 12 , y = k 2 / 12 and z = x + y , ∑ i = 0 8 ψ i represents the 8 nearest neighboring points and a centre point in the computational domain.
Equation (18) is simplified using Equations (10)-(13) and then r = e π ξ with θ = π η is applied. Then, we can arrive at the following equation in terms of ξ and η .
∂ 2 ω ∂ ξ 2 + ∂ 2 ω ∂ η 2 + c ∂ ω ∂ ξ + d ∂ ω ∂ η = F (41)
where the coefficients c i , j , d i , j and F i , j are
c = − R e 2 ( ∂ ψ ∂ η ) (42)
d = R e 2 ( ∂ ψ ∂ ξ ) (43)
and
F = − e − 2 π ξ 4 π 2 R e ⋅ R m ⋅ β [ − ∂ ψ ∂ η ∂ A ∂ η ∂ 2 A ∂ ξ 2 − ∂ ψ ∂ ξ ∂ A ∂ ξ ∂ 2 A ∂ η 2 + 2 π ∂ ψ ∂ η ∂ A ∂ η ∂ A ∂ ξ + ∂ A ∂ ξ ∂ ψ ∂ η ∂ 2 A ∂ ξ ∂ η + ∂ ψ ∂ ξ ∂ A ∂ η ∂ 2 A ∂ ξ ∂ η − 2 ∂ A ∂ ξ ∂ A ∂ η ∂ 2 ψ ∂ ξ ∂ η − 2 π ∂ ψ ∂ ξ ( ∂ A ∂ η ) 2 + ( ∂ A ∂ η ) 2 ∂ 2 ψ ∂ ξ 2 + ( ∂ A ∂ ξ ) 2 ∂ 2 ψ ∂ η 2 ] (44)
On substitution of Equations (21)-(24) in the vorticity Equation (41) we get the discretized form as below.
D ξ 2 ω i , j + D η 2 ω i , j + c i , j D ξ ω i , j + d i , j D η ω i , j + τ i , j = F i , j (45)
and the truncation error τ in (45) is
τ i , j = − ( c h 2 6 ∂ 3 ω ∂ ξ 3 + h 2 12 ∂ 4 ω ∂ ξ 4 + d k 2 6 ∂ 3 ω ∂ η 3 + k 2 12 ∂ 4 ω ∂ η 4 ) i , j + O ( h 4 , k 4 ) (46)
Now, to eliminate the third and fourth derivatives of ω that are present in the truncation error term, we differentiate (41) once and twice with respect to ξ and η respectively, to yield the following.
∂ 3 ω ∂ ξ 3 = − ∂ 3 ω ∂ ξ ∂ η 2 − c ∂ 2 ω ∂ ξ 2 − d ∂ 2 ω ∂ ξ ∂ η − ∂ c ∂ ξ ∂ ω ∂ ξ − ∂ d ∂ ξ ∂ ω ∂ η − ∂ F ∂ ξ (47)
∂ 4 ω ∂ ξ 4 = − ∂ 4 ω ∂ ξ 2 ∂ η 2 + c ∂ 3 ω ∂ ξ ∂ η 2 − d ∂ 3 ω ∂ ξ 2 ∂ η + ( c 2 − 2 ∂ c ∂ ξ ) ∂ 2 ω ∂ ξ 2 + ( c d − 2 ∂ d ∂ ξ ) ∂ 2 ω ∂ ξ ∂ η + ( c ∂ c ∂ ξ − ∂ 2 c ∂ ξ 2 ) ∂ ω ∂ ξ + ( c ∂ d ∂ ξ − ∂ 2 d ∂ ξ 2 ) ∂ ω ∂ η + c ∂ F ∂ ξ − ∂ 2 F ∂ ξ 2 (48)
∂ 3 ω ∂ η 3 = − ∂ 3 ω ∂ ξ 2 ∂ η − c ∂ 2 ω ∂ ξ ∂ η − d ∂ 2 ω ∂ η 2 − ∂ c ∂ η ∂ ω ∂ ξ − ∂ d ∂ η ∂ ω ∂ η − ∂ F ∂ η (49)
∂ 4 ω ∂ η 4 = − ∂ 4 ω ∂ ξ 2 ∂ η 2 − c ∂ 3 ω ∂ ξ ∂ η 2 + d ∂ 3 ω ∂ ξ 2 ∂ η + ( c d − 2 ∂ c ∂ η ) ∂ 2 ω ∂ ξ ∂ η + ( d 2 − 2 ∂ d ∂ η ) ∂ 2 ω ∂ η 2 + ( d ∂ c ∂ η − ∂ 2 c ∂ η 2 ) ∂ ω ∂ ξ + ( d ∂ d ∂ η − ∂ 2 d ∂ η 2 ) ∂ ω ∂ η + d ∂ F ∂ η − ∂ 2 F ∂ η 2 (50)
Now we substitute (42)-(50) in (45) and then apply the D-operators from (25)-(32). Then, the resulting equation is simplified to arrive at the fourth order accurate representation of vorticity Equation (41) as below.
− 8 ( e k 2 + f h 2 − 2 z ) ω 0 + ( 4 k 2 e + 2 h k 2 g − 8 z − 4 h z c ) ω 1 + ( 4 h 2 f + 2 h 2 k o − 8 z − 4 k z d ) ω 2 + ( 4 k 2 e − 2 h k 2 g − 8 z + 4 h z c ) ω 3 + ( 4 h 2 f − 2 h 2 k o − 8 z + 4 k z d ) ω 4 + ( 4 z + 2 h c z + 2 k d z + h k l ) ω 5 + ( 4 z − 2 h c z + 2 k d z − h k l ) ω 6 + ( 4 z − 2 h c z − 2 k d z + h k l ) ω 7 + ( 4 z + 2 h c z − 2 k d z − h k l ) ω 8 = − h 2 k 2 R e R m β e − 2 π ξ π 2 { G + ( x c D ξ + y d D η ) [ G ] + ( x D ξ 2 + y D η 2 ) [ G ] } (51)
where
G = − D η [ ψ ] D η D ξ 2 [ A ] − D ξ [ ψ ] D ξ D η 2 [ A ] + 2 π D η [ ψ ] D η D ξ [ A ] + D ξ [ A ] D η [ ψ ] D ξ D η [ A ] + D ξ [ ψ ] D η [ A ] D ξ D η [ A ] − 2 D ξ [ A ] D η [ A ] D ξ D η [ ψ ] − 2 π D ξ [ ψ ] D η [ A ] D η [ A ] + D η [ A ] D η [ A ] D ξ 2 [ ψ ] + D ξ [ A ] D ξ [ A ] D η 2 [ ψ ] (52)
where the values of ψ around ψ i , j are indexed as shown in the 2D compact stencil (
e = 1 + x { c 2 − R e D ξ D η [ ψ ] } (53)
f = 1 + y { d 2 + R e D ξ D η [ ψ ] } (54)
g = − R e 2 [ { D η + y d D η 2 + x c D ξ D η + z D ξ 2 D η } [ ψ ] + y π 2 e 2 π ξ D η [ ω ] ] (55)
o = R e 2 [ { D ξ + x c D ξ 2 + y d D ξ D η + z D ξ D η 2 } [ ψ ] + x π 2 e 2 π ξ D ξ [ ω ] ] (56)
l = z c d + R e { x D ξ 2 − y D η 2 } [ ψ ] (57)
In addition to the above, we ill represent Equations ((42) and (43)) in fourth order accurate discrete form as below
c = − R e { 1 2 D η [ ψ ] + y D ξ 2 D η [ ψ ] + y π 2 e 2 π ξ D η [ ω ] } (58)
d = R e { 1 2 D ξ [ ψ ] + x D ξ D η 2 [ ψ ] + x π 2 e 2 π ξ D ξ [ ω ] } (59)
Similar to the case of streamfunction equation, first the equation for magnetic field (19) is expressed in terms of magnetic stream function using Equations ((12) and (13)) and the velocities are replaced using (10) and (11), so that we get the magnetic streamfunction equation as given below
∂ 2 A ∂ ξ 2 + ∂ 2 A ∂ η 2 + C ∂ A ∂ ξ + D ∂ A ∂ η = 0 (60)
where the coefficients C and D are
C = − R m 2 ( ∂ ψ ∂ η ) (61)
D = R m 2 ( ∂ ψ ∂ ξ ) (62)
Substituting (21)-(24) in the magnetic streamfunction Equation (60), we get
D ξ 2 A i , j + D η 2 A i , j + C i , j D ξ A i , j + D i , j D η A i , j + ζ i , j = 0 (63)
and the truncation error in the Equation (63) is
ζ i , j = − ( C h 2 6 ∂ 3 A ∂ ξ 3 + h 2 12 ∂ 4 A ∂ ξ 4 + D k 2 6 ∂ 3 A ∂ η 3 + k 2 12 ∂ 4 A ∂ η 4 ) i , j + O ( h 4 , k 4 ) (64)
It may be noted that the coefficients in the magnetic streamfunction equations are denoted by C and D while those in vorticity equation are denoted by c and d. Also, the coefficient D in the above equation may not confuse with the operators like D ξ because operators will always have a suffix ξ or η . Elimination of higher order derivatives of A in (64) is done by differentiating the magnetic streamfunction Equation (60) once and twice with respect to ξ and η respectively, to yield the following.
∂ 3 A ∂ ξ 3 = − ∂ 3 A ∂ ξ ∂ η 2 − C ∂ 2 A ∂ ξ 2 − D ∂ 2 A ∂ ξ ∂ η − ∂ C ∂ ξ ∂ A ∂ ξ − ∂ D ∂ ξ ∂ A ∂ η (65)
∂ 4 A ∂ ξ 4 = − ∂ 4 A ∂ ξ 2 ∂ η 2 + C ∂ 3 A ∂ ξ ∂ η 2 − D ∂ 3 A ∂ ξ 2 ∂ η + ( C 2 − 2 ∂ C ∂ ξ ) ∂ 2 A ∂ ξ 2 + ( C D − 2 ∂ D ∂ ξ ) ∂ 2 A ∂ ξ ∂ η + ( C ∂ C ∂ ξ − ∂ 2 C ∂ ξ 2 ) ∂ A ∂ ξ + ( C ∂ D ∂ ξ − ∂ 2 D ∂ ξ 2 ) ∂ A ∂ η (66)
∂ 3 A ∂ η 3 = − ∂ 3 A ∂ ξ 2 ∂ η − C ∂ 2 A ∂ ξ ∂ η − D ∂ 2 A ∂ η 2 − ∂ C ∂ η ∂ A ∂ ξ − ∂ D ∂ η ∂ A ∂ η (67)
∂ 4 A ∂ η 4 = − ∂ 4 A ∂ ξ 2 ∂ η 2 − C ∂ 3 A ∂ ξ ∂ η 2 + D ∂ 3 A ∂ ξ 2 ∂ η + ( C D − 2 ∂ C ∂ η ) ∂ 2 A ∂ ξ ∂ η + ( D 2 − 2 ∂ D ∂ η ) ∂ 2 A ∂ η 2 + ( D ∂ C ∂ η − ∂ 2 C ∂ η 2 ) ∂ A ∂ ξ + ( D ∂ D ∂ η − ∂ 2 D ∂ η 2 ) ∂ A ∂ η (68)
Now we first substitute (61)-(68) in Equation (63) and then apply D-operators from Equations (25)-(32) to arrive at the fourth order accurate discretized representation of the magnetic streamfunction Equation (60) as given below.
[ − α D ξ 2 − β ′ D η 2 + γ D ξ + λ D η + μ D ξ D η ] A − ( x + y ) [ D ξ 2 D η 2 − C D ξ D η 2 − D D ξ 2 D η ] A = 0 (69)
where the coefficients α , β ′ , γ , λ , μ , C and D appearing above are given by
α = ( 1 + x C 2 ) − [ R m x D ξ D η ] ψ (70)
β ′ = ( 1 + y D 2 ) − [ R m y D η 2 ] ψ (71)
γ = C + R m 2 x [ D ξ 2 D η − C D ξ D η ] ψ − R m 2 y [ D ξ 2 D η − D D η 2 ] ψ − [ R m 2 y π 2 e 2 π ξ d η ] ω (72)
λ = D + R m 2 x [ D ξ D η 2 + C D ξ 2 ] ψ − R m 2 y [ D ξ D η 2 + D D ξ D η ] ψ + [ R m 2 x π 2 e 2 π ξ ( 2 π + D ξ ) ] ω (73)
μ = R m [ − x D ξ 2 + y D η 2 ] ψ − ( x + y ) C D (74)
In addition to the above, we ill represent Equations ((61) and (62)) in fourth order accurate discrete form as below
C = R m 2 [ D η + 2 y D ξ 2 D η ] ψ + [ R m y π 2 e 2 π ξ D η ] ω (75)
D = − R m 2 [ D ξ + 2 x D ξ D η 2 ] ψ − [ R m x π 2 e 2 π ξ ( 2 π + D ξ ) ] ω (76)
In Equation (20) velocities are eliminated using (10) and (11) then, r = e π ξ and θ = π η transformations are employed. This leads to the following equation for temperature.
∂ 2 T ∂ ξ 2 + ∂ 2 T ∂ η 2 + c ′ ∂ T ∂ ξ + d ′ ∂ T ∂ η = 0 (77)
where the coefficients, c ′ and d ′ are given by
c ′ = − R e P r 2 ( ∂ ψ ∂ η ) (78)
d ′ = R e P r 2 ( ∂ ψ ∂ ξ ) (79)
Equations (21)-(24) are applied on the above equation so that we have
D ξ 2 T i , j + D η 2 T i , j + c ′ i , j D ξ T i , j + d ′ i , j D η T i , j + γ i , j = 0 (80)
The truncation error in the Equation (80) is
γ i , j = − ( c ′ h 2 6 ∂ 3 T ∂ ξ 3 + h 2 12 ∂ 4 T ∂ ξ 4 + d ′ k 2 6 ∂ 3 T ∂ η 3 + k 2 12 ∂ 4 T ∂ η 4 ) + O ( h 4 , k 4 ) (81)
In order to higher derivatives of T present in the truncation error terms, we differentiate the energy Equation (77) once and twice with respect to ξ and η respectively and we get the following
∂ 3 T ∂ ξ 3 = − ∂ 3 T ∂ ξ ∂ η 2 − c ′ ∂ 2 T ∂ ξ 2 − d ′ ∂ 2 T ∂ ξ ∂ η − ∂ c ′ ∂ ξ ∂ T ∂ ξ − ∂ d ′ ∂ ξ ∂ T ∂ η (82)
∂ 4 T ∂ ξ 4 = − ∂ 4 T ∂ ξ 2 ∂ η 2 + c ′ ∂ 3 T ∂ ξ ∂ η 2 − d ′ ∂ 3 T ∂ ξ 2 ∂ η + ( c ′ 2 − 2 ∂ c ′ ∂ ξ ) ∂ 2 T ∂ ξ 2 + ( c ′ d ′ − 2 ∂ d ′ ∂ ξ ) ∂ 2 T ∂ ξ ∂ η + ( c ′ ∂ c ′ ∂ ξ − ∂ 2 c ′ ∂ ξ 2 ) ∂ T ∂ ξ + ( c ′ ∂ d ′ ∂ ξ − ∂ 2 d ′ ∂ ξ 2 ) ∂ T ∂ η (83)
∂ 3 T ∂ η 3 = − ∂ 3 T ∂ ξ 2 ∂ η − c ′ ∂ 2 T ∂ ξ ∂ η − d ′ ∂ 2 T ∂ η 2 − ∂ c ′ ∂ η ∂ T ∂ ξ − ∂ d ′ ∂ η ∂ T ∂ η (84)
∂ 4 T ∂ η 4 = − ∂ 4 T ∂ ξ 2 ∂ η 2 − c ′ ∂ 3 T ∂ ξ ∂ η 2 + d ′ ∂ 3 T ∂ ξ 2 ∂ η + ( d ′ 2 − 2 ∂ d ′ ∂ η ) ∂ 2 T ∂ η 2 + ( c ′ d ′ − 2 ∂ c ′ ∂ η ) ∂ 2 T ∂ ξ ∂ η + ( d ′ ∂ c ′ ∂ η − ∂ 2 c ′ ∂ η 2 ) ∂ T ∂ ξ + ( d ′ ∂ d ′ ∂ η − ∂ 2 d ′ ∂ η 2 ) ∂ T ∂ η (85)
Now substituting Equations (78)-(85) in the truncation error terms of Equation (77), and applying operator D to the derivatives, we will get the fourth order discretized form of energy equation, as given below
[ − ς D ξ 2 − ρ D η 2 + ϖ D ξ + ϑ D η + ε D ξ D η ] T − ( x + y ) [ D ξ 2 D η 2 − C D ξ D η 2 − D D ξ 2 D η ] T = 0 (86)
where the coefficients ς , ρ , ϖ , ϑ , ε , c ′ and d ′ appearing above are given by
ς = ( 1 + x c ′ 2 ) − [ R e P r x D ξ D η ] ψ (87)
ρ = ( 1 + y d ′ 2 ) − [ R e P r y D η 2 ] ψ (88)
ϖ = c ′ + R e P r 2 x [ D ξ 2 D η − c ′ D ξ D η ] ψ − R e P r 2 y [ D ξ 2 D η − d ′ D η 2 ] ψ − [ R e P r 2 y π 2 e 2 π ξ d η ] ω (89)
ϑ = d ′ + R e P r 2 x [ D ξ D η 2 + c ′ D ξ 2 ] ψ − R e P r 2 y [ D ξ D η 2 + d ′ D ξ D η ] ψ + [ R e P r 2 x π 2 e 2 π ξ ( 2 π + D ξ ) ] ω (90)
ε = R e P r [ − x D ξ 2 + y D η 2 ] ψ − ( x + y ) c ′ d ′ (91)
In addition to the above, we will represent Equations ((78) and (79)) in fourth order accurate discrete form as below
c ′ = R e P r 2 [ D η + 2 y D ξ 2 D η ] ψ + [ R e P r y π 2 e 2 π ξ D η ] ω (92)
d ′ = − R e P r 2 [ D ξ + 2 x D ξ D η 2 ] ψ − [ R e P r x π 2 e 2 π ξ ( 2 π + D ξ ) ] ω (93)
We evaluate the performance of the above discretization scheme by running numerical experiments for the MHD flow around a cylinder, since it is one of the classic problems in MHD flow where the flow features have been studied by many authors. Hence we can compare our high accuracy results and discuss on the performance of proposed discretization scheme. In order to cluster more grid points near the surface of the cylinder, an exponential transformation is employed. It is also assumed that the magnetic field will not penetrate across the boundary of the cylinder. The boundary conditions used are the following. On the surface of the cylinder, ψ = 0 , ∂ ψ / ∂ r = 0 , ω = ∂ 2 ψ / ∂ r 2 and A = 0 . Far away from the cylinder which is denoted by ξ → ∞ , the imposed conditions are ψ ~ r s i n ( θ ) , ω → 0 , A → r s i n ( θ ) . The axis of symmetry is given by η = 0 and η = 1 , on which lines, the boundary conditions imposed are ψ = 0 , ω = 0 , and A = 0 . For solving the energy equation, we impose a uniform temperature T = 1 on the cylinder surface and T = 0 at far-field distances. On the axis of symmetry, ∂ T / ∂ η = 0 is imposed. The derivatives appearing in the boundary conditions for streamfunction vorticity and temperature are
∂ ψ ∂ ξ = 0 (94)
∂ 2 ψ ∂ ξ 2 = − π 2 ω 2 (95)
∂ T ∂ η = 0 (96)
The above derivatives are replaced with suitable one-sided fourth order accurate finite differences as given below
ψ 1 , j = ( 48 ψ 2 , j − 36 ψ 3 , j + 16 ψ 4 , j − 3 ψ 5 , j ) 25 (97)
ψ n + 1 , j = − ( 48 ψ n , j − 36 ψ n − 1 , j + 16 ψ n − 2 , j − 3 ψ n − 3 , j ) 25 (98)
T i , 1 = ( 48 T i , 2 − 36 T i , 3 + 16 T i , 4 − 3 T i , 5 ) 25 (99)
T i , m + 1 = − ( 48 T i , m − 36 T i , m − 1 + 16 T i , m − 2 − 3 T i , m − 3 ) 25 (100)
The vorticity boundary condition used on the surface of the cylinder is the Briley’s formula
ω 1 , j = − ( 108 ψ 2 , j − 27 ψ 3 , j + 4 ψ 4 , j ) 18 h 2 (101)
The algebraic system of Equations ((40), (51), (69) and (86)) exhibits diagonal dominance and hence the Gauss-Seidel iterative procedure is used to solve the discretized coupled equations. The convergence criterion is set as the norm of dynamic residuals and computations are terminated if the norm is less than 10−8. An additional difficulty faced due to the high accuracy numerical scheme is that, at every iteration, before the Equation (51) can be solved, the values of G , D ξ [ G ] , D η [ G ] , D ξ 2 [ G ] , D η 2 [ G ] and all coefficients therein, Equations (53)-(58) should be evaluated at the required grid points. This same is true while solving for magnetic stream function, A and the temperature T equations. It is also observed that the presence of several nonlinearities on the right side of (51) which makes the numerical solution time consuming and parameter restricted. This means a higher computational cost in terms of memory is required, especially at high resolution grids like 512 × 512 . In addition, it takes more time for convergence when compared to the use of traditional unwinding schemes. The lift and drag are the two forces acting on the cylinder due to fluid flow. Because of the two counter rotating symmetric vortices behind the circular cylinder, the lift forces get canceled. The drag coefficients (pressure, viscous and total) are calculated using the relations
C P = 4 π R e ∫ 0 1 ( ∂ ω ∂ ξ ) ξ = 0 s i n ( π η ) d η (102)
C V = − 4 π R e ∫ 0 1 ω ξ = 0 s i n ( π η ) d η (103)
C D = C P + C V (104)
The heat flux q ( θ ) , local Nusselt number N u and mean Nusselt number N m are calculated using
q ( θ ) = k ( T s − T ∞ ) a ( ∂ T ∂ ξ ) ξ = 0 (105)
N u ( θ ) = 2 a q ( θ ) k ( T s − T ∞ ) = − 2 π ( ∂ T ∂ ξ ) ξ = 0 (106)
N m = ∫ 0 1 ( ∂ T ∂ ξ ) ξ = 0 d η (107)
In the above, all derivatives are approximated with fourth order accurate central differences. However, for points near the boundary, appropriate forward or backward fourth order finite difference expressions are used. Since fine grid solutions are considered, the integral in the above is carried out using the Simpson’s rule.
We have computed the fourth order accurate results for R e = 40 with magnetic Reynolds number ranging 0 ≤ R m ≤ 2 for different values of the magnetic field strengths β . The values of Prandtl number used in the present study are P r = 0.065 , 0.73, 5 and 7 which will typically represent liquids such as liquid lithium, air, aqueous KOH solution or some refrigerants and saline water. The numerical results are presented using the one obtained from the finest grid 256 × 256 , while the coarser grids employed are 128 × 128 and 64 × 64 . The convergence criterion used in this study is that the norm of the dynamic residuals ε < 10 − 8 . The far-field distance or the computational domain size is chosen as 41 times the radius of cylinder, which is a fairly a large space to acquire accurate results.
Re | 64 × 64 | 128 × 128 | 256 × 256 | ||
---|---|---|---|---|---|
10 | 0.1 | 2.0 | 2.908 | 2.907 | 2.906 |
0.5 | 2.0 | 3.464 | 3.462 | 3.461 | |
1 | 0.1 | 2.784 | 2.785 | 2.785 | |
1.5 | 0.1 | 2.754 | 2.756 | 2.757 | |
2 | 0.1 | 2.732 | 2.736 | 2.737 | |
20 | 0.1 | 2.0 | 2.068 | 2.070 | 2.069 |
0.5 | 2.0 | 2.384 | 2.391 | 2.391 | |
1 | 0.1 | 1.995 | 1.999 | 2.000 | |
1.5 | 0.1 | 1.971 | 1.976 | 1.977 | |
2 | 0.1 | 1.952 | 1.959 | 1.960 | |
40 | 0.1 | 2.0 | 1.531 | 1.540 | 1.541 |
0.5 | 2.0 | 1.692 | 1.731 | 1.734 | |
1 | 0.1 | 1.475 | 1.488 | 1.489 | |
1.5 | 0.1 | 1.452 | 1.469 | 1.471 | |
2 | 0.1 | 1.434 | 1.454 | 1.456 |
FLOW PARAMETERS | |||||
---|---|---|---|---|---|
Present study (4th order) | 1.017 | 0.532 | 1.549 | 5.47 | 53.4 |
Sekhar et al. [ | 0.99 | 0.51 | 1.5 | 5.57 | 54 |
Tuann & Olson [ | 1.095 | 0.58 | 1.675 | 5.2 | 54.8 |
Bharti et al. [ | 0.9976 | 0.5316 | 1.5292 | 5.45 | 53.71 |
Kim et al. [ | --- | --- | 1.51 | --- | --- |
Sanyasiraju & Manjula [ | --- | --- | 1.53 | 5.21 | 51.025 |
Raju & Sekhar [ | --- | --- | 1.51 | 5.383 | 52.734 |
Takami and Keller [ | --- | --- | 1.536 | 5.65 | 53.55 |
Dennis and Chang [ | 0.998 | 0.524 | 1.522 | 5.69 | 53.8 |
Collins & Dennis [ | --- | --- | --- | 5.3 | 53.6 |
Nieuwstadt and Keller [ | --- | --- | 1.55 | 5.357 | 53.34 |
Coutanceau and Bouard [ | --- | --- | --- | 5.26 | 53.5 |
Fornberg [ | --- | --- | 1.498 | 5.48 | 51.5 |
Rogers & Kwak [ | 1.011 | 0.538 | 1.549 | 5.58 | 53.0 |
He and Doolen [ | --- | --- | 1.499 | 5.49 | 52.84 |
Park et al. [ | 0.99 | 0.52 | 1.51 | --- | --- |
Ye et al. [ | --- | --- | 1.52 | 5.54 | --- |
Niu et al. [ | --- | --- | 1.574 | 5.51 | 53.8 |
Silva et al. [ | --- | --- | 1.54 | --- | --- |
Tseng and Ferziger [ | --- | --- | 1.53 | 5.42 | --- |
Chakraborty et al. [ | 0.9943 | 0.5229 | 1.5172 | 5.21 | 53.6 |
Dinget al. [ | --- | --- | 1.713 | 5.4 | 53.5 |
Shu et al. [ | --- | --- | --- | 5.40 | --- |
Udhayakumar et al. [ | --- | --- | 1.583 | 5.338 | 53.43 |
Udhayakumar et al. [ | --- | --- | 1.596 | 5.387 | 53.88 |
Udhayakumar et al. [ | --- | --- | 1.598 | 5.382 | 53.85 |
R e and for three different grids in presence of magnetic field. From the table, it is noted that 128 × 128 grid is found to be optimum. Further refining of grids, that is employing 512 × 512 , etc. will not give more accuracy because the accuracy of the numerical scheme is decided by method of discretization and not by the fineness of grid.
Literature Reports | Nm |
---|---|
Present study (4th order) | 3.290 |
Bharti et al. [ | 3.282 |
Udhayakumar et al. [ | 3.285 |
Udhayakumar et al. [ | 3.294 |
Udhayakumar et al. [ | 3.306 |
Dennis et al. [ | 3.480 |
Apelt and Ledwich [ | 3.255 |
Badr [ | 3.480 |
Jafroudi and Yang [ | 3.200 |
Chen et al. [ | 3.470 |
Aldoss et al. [ | 3.480 |
Lange [ | 3.280 |
Sparrow [ | 3.082 |
Soareset al. [ | 3.200 |
Biswas and Sandip Sarkar [ | 3.253 |
Soareset al. [ | 3.190 |
Vimala et al. [ | 3.313 |
present work which is attributed to the higher order numerical scheme employed in contrast to traditional first or second order accurate data in the literature. Further the data for mean Nusselt number N m is also compared against other reported data [
The response of separation length ( l s ) and the separation angle ( θ s ) of the recirculation bubble with the Alfvén number ( β ) is presented in
is required for the same level of wake suppression. The Lorentz force will directly change the fluid velocity and according to Equation (2) and hence the vorticity of the fluid changes due to applied magnetic field. Together with a gradual decrease in the length of the standing vortex near the cylinder, a growing region of positive vortices are seen when the magnetic field strength is increased (not shown) and this is also predicted theoretically [
The dependence of pressure drag coefficient C P , viscous drag coefficient C V , and total drag coefficient C D on β and R m are shown in
The forced convective heat transfer in the fluid flow is studied for R e = 40 with 0 ≤ R m ≤ 2 , 0 ≤ β ≤ 50 for Prandtl numbers 0.065, 0.73, 5, 7 which may represent typical fluids like liquid metals, air, KOH and water respectively. The surface Nusselt number on the surface of the cylinder and its effect on the magnetic field is shown in
Rm | β | CP | CP [ | CV | CV [ | CD | CD [ | lS | lS [ | θS | θS [ |
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0 | 1.017 | 0.99 | 0.532 | 0.51 | 1.549 | 1.50 | 5.47 | 5.57 | 53.4 | 54 |
0.5 | 0.4 | 0.947 | 0.94 | 0.505 | 0.50 | 1.452 | 1.40 | 3.23 | 4.5 | 47.8 | 52.5 |
0.5 | 0.8 | 0.991 | 0.91 | 0.522 | 0.485 | 1.514 | 1.405 | 2.25 | 3.25 | 41.5 | 48 |
0.5 | 1 | 1.023 | 0.90 | 0.535 | 0.48 | 1.558 | 1.38 | 2.04 | 2.9 | 39.4 | 46.5 |
0.5 | 4 | 1.342 | 1.11 | 0.642 | 0.57 | 1.984 | 1.68 | 1.16 | 1.5 | 17.6 | 29 |
0.5 | 6 | 1.489 | 1.42 | 0.688 | 0.60 | 2.177 | 2.02 | 1.00 | 1.25 | 4.9 | 22 |
1 | 0.4 | 0.884 | 0.93 | 0.479 | 0.48 | 1.363 | 1.41 | 2.49 | 4.1 | 43.6 | 51.5 |
1 | 0.8 | 1.033 | 0.86 | 0.536 | 0.46 | 1.570 | 1.32 | 1.57 | 2.55 | 31.6 | 44.5 |
1 | 1 | 1.098 | 0.87 | 0.56 | 0.47 | 1.658 | 1.34 | 1.43 | 2.12 | 28.1 | 40.8 |
1 | 3 | 1.472 | 1.17 | 0.681 | 0.59 | 2.153 | 1.76 | 1.00 | 1.25 | 28.1 | 20.1 |
2 | 0.2 | 0.887 | 0.95 | 0.482 | 0.51 | 1.369 | 1.46 | 3.32 | 4.95 | 48.5 | 52.8 |
2 | 0.4 | 0.794 | 0.90 | 0.441 | 0.43 | 1.235 | 1.33 | 1.86 | 3.83 | 37.3 | 50.9 |
2 | 0.8 | 1.139 | 0.78 | 0.568 | 0.44 | 1.708 | 1.22 | 1.11 | 1.95 | 14.8 | 39.7 |
2 | 1 | 1.244 | 0.77 | 0.618 | 0.43 | 1.906 | 1.21 | 1.05 | 1.49 | 8.4 | 30.5 |
which is the rear stagnation point (red, black and blue lines). That is, along with suppression of flow separation, the enhancement of heat transfer is also suppressed. These effects proportionally increase for fluid with higher Prandtl numbers. The application of external magnetic field causes the thickness of the viscous boundary layer to increase. The maximum heat transfer region lies in the front stagnation point. At this point the magnetic field casus a decrease in Nu. The effect of electrical conductivity on the Nu is shown in
non-monotonic behavior takes place.
The variation of temperature with radial distance from the surface of the cylinder is plotted in the
larger is the density of isotherm lines in the upstream region, which means that the thermal boundary layer thickness is reduced. This is not the case when the magnetic field strength is varied for a fixed P r , where the thermal boundary layer thickness does not significantly change.
In order to evaluate the accuracy of our numerical results, we have taken the
drag coefficient values C D from different grids of grid step size h. Then, its divided difference with respect to h is calculated as
Δ ( C D ) d h = ( C D ) h 1 − ( C D ) h 2 h 1 − h 2 = y (108)
A plot between y and the grid step size h is made on a log-log scale which is shown in
In this paper, we have studied the problem of electrically conducting flow over a circular cylinder in the presence a steady magnetic field using a high accuracy finite difference scheme. When compared to the results obtained using second order mixed with upwind schemes, it is found that relatively lower magnetic field is sufficient for suppressing the flow separation. In the present study, it is found that the suppression of flow separation depends on both conductivity of the fluid and the strength of magnetic field, while the study using the low R m approximation reveals that the separation is nearly independent of conductivity of the fluid. For small values of magnetic Reynolds number R m the total drag coefficient decreases when β ≈ 0.5 and then the drag coefficient increases for higher values of β . In the absence of external magnetic field, the minimum heat transfer takes place at the separation point. When magnetic field is incre- ased, the minimum of the Nusselt number N u is shifted towards the rear stagnation point. The mean Nusselt number ( N m ) depends non-monotonically with the electrical conductivity ( R m ) of the fluid in the region 0.4 ≤ β ≤ 0.65 . The degradation of heat transfer by the application of external magnetic field up to the values of β ≈ 0.5 , further there is an increase of heat transfer is observed. The temperature gradient significantly increases with the increase of magnetic field only near the cylinder body. High thermal gradients occur in fluids with higher P r leading to higher heat transfer. The overall accuracy of the derived numerical scheme is estimated to be four using divided difference principle.
One of the authors, R.Sivakumar thank UGC for supporting this work vide major research project grant letter F. No. 37-312/2009 (SR) dated January 12, 2010, and also DST for their FIST funding vide order SR/FST/PSII-021/2009 dated August 13, 2010 towards establishing a basic computing cluster facility. Finally Abin Rejeesh would like to thank DST for providing INSPIRE fellowship vide letter No. DST/INSPIRE Fellowship/2011/[
Abin Rejeesh, A.D., Udhayakumar, S., Sekhar, T.V.S. and Sivakumar, R. (2017) Prediction of Better Flow Control Parameters in MHD Flows Using a High Accuracy Finite Difference Scheme. American Journal of Computational Mathematics, 7, 243-275. https://doi.org/10.4236/ajcm.2017.73021
A = Magnetic stream function.
a , d = Radius and diameter of cylinder.
c = Specific heat capacity of fluid.
C P , C V , C D = Pressure, viscous and total drag coefficient.
E = Electric field.
Η , H ∞ = Induced Magnetic field and applied far field magnetic field.
h r , h θ = Radial and transverse magnetic field components.
J = Electric current density.
l s = Separation length of the wake.
N = Interaction parameter or Stuart number = a σ H ∞ 2 / ( ρ U ∞ ) .
N u = Local Nusselt number:
N m = Average Nusselt number
p = Dimensionless pressure.
P r = Prandtl number = ν / κ .
q = heat flux.
q r , q θ = Radial and transverse velocity components.
R e = Reynolds number = 2 a U ∞ / ν
R m = Magnetic Reynolds number = 2 a U ∞ μ m σ
T = Dimensionless temperature.
T s = Surface temperature of the cylinder.
T ∞ = Far field free stream temperature.
U ∞ = Uniform free stream velocity in the far field:
α = Thermal expansion coefficient of the fluid:
β = Alfven number.
δ T = Thermal boundary layer thickness:
κ = Thermal diffusivity of the fluid:
μ m = Magnetic permeability of the fluid.
μ = Dynamic viscosity of the fluid:
ν = Kinematic viscosity of the fluid = μ / ρ
ρ = Density of the fluid:
σ = Electrical conductivity of the fluid:
θ s = Separation angle of the wake.
( r , θ ) = 2D cylindrical polar coordinates:
( ξ , η ) = Coordinates after r = e π ξ , and θ = π η transformation.
( ψ , ω ) = Streamfunction and vorticity.