A New Approximate Analytical Solutions for Two-and Three-Dimensional Unsteady Viscous Incompressible Flows by Using the Kinetically Reduced Local Navier-Stokes Equations

In this work, the kinetically reduced local Navier-Stokes equations are applied to the simulation of twoand three-dimensional unsteady viscous incompressible flow problems. The reduced differential transform method is used to find the new approximate analytical solutions of these flow problems. The new technique has been tested by using four selected multidimensional unsteady flow problems: twoand three-dimensional Taylor decaying vortices flow, Kovasznay flow, and three-dimensional Beltrami flow. The convergence analysis was discussed for this approach. The numerical results obtained by this approach are compared with other results that are available in previous works. Our results show that this method is efficient to provide new approximate analytic solutions. Moreover, we found that it has highly precise solutions with good convergence, less time consuming, being easily implemented for high Reynolds numbers, and low Mach numbers.


Introduction
Many of the physical phenomena in fluid mechanics are formulated according to the unsteady viscous incompressible Navier-Stokes (INS) equations, which has the nondimensional formula consisting of the momentum equations and the continuity equation [1][2][3][4][5][6][7][8] where  is the physical time, u is the velocity field,  is the pressure, and  is the Reynolds number ( = U/], where U is the scale velocity field,  is the characteristic length, and ] is the kinematic viscosity of the fluid).Analytical and numerical solutions of INS equations are known difficulty because they are non-linear equations, and they do not find the time evolution equation for the pressure that must be determined by solving the Passion equation at each time step, which requires effort and time.Therefore, there are a lot of studies that have developed an alternative formula description of incompressible fluid flows.One of these alternative formulas is the kinetically reduced local Navier-Stokes (KRLNS) equations which was suggested in [1] for the thermodynamic description of incompressible fluid flows at low Mach numbers.The system of KRLNS equations is such that where  is the Mach number (  = U/  is the ratio of the characteristic flow speed U to the isentropic sound speed   ), and  is the grand potential.The time scale in INS equations related to that of KRLNS equations;   () =  ×   .
All studies which have presented the KRLNS equations for simulation of unsteady incompressible viscous flow problems, used the numerical schemes for solving these equations.The KRLNS equations are proposed for the simulation of low Mach number flows in [2], and used the spectral element method to find the numerical solution of the three-dimensional Taylor Green vortex flow.In [3], two-dimensional KRLNS system is simplified and compared with a Chorin's artificial compressibility method for steady state computation of the flow in a lid-driven cavity at various Reynolds numbers, the Taylor Green vortex flow is demonstrated that the KRLNS equations correctly describe the time evolution of the velocity and of the pressure, for this purpose, the explicit Mac Cormack scheme is used.In [5] the KRLNS equations are applied to two-dimensional simulation of doubly periodic shear layers and decaying homogeneous isotropic turbulence, to solve these equations have been used the central difference scheme for the spatial discretization in both advection and diffusion terms and four stages Runge-Kutta method for the time integration, the numerical results are compared with those obtained by the artificial compressibility method, the lattice Boltzmann method, and the pseudospectral method.Higher order difference approximations are used in [6] to find the solutions of the KRLNS equations which are applied for two-dimensional simulations of Womersley problem and doubly periodic shear layers.
The main purpose of this paper is to find new approximate analytical solutions for two-and three-dimensional unsteady viscous incompressible flow problems.To achieve this objective, the flow problems that are described by alternative formulas of Navier-stokes equations, which are named KRLNS (3) and (4), the reduced differential transform method (RDTM) is proposed.The reasons that encourage us to propose RDTM to solve the present problems are being an effective and efficient method to find approximate analytical solutions for nonlinear equations and we believe that it has been achieved for the first time in its study.Moreover, we extend the application of RDTM and compare its reliability and efficiency with other methods.New approximate analytical solutions for two-and three-dimensional unsteady viscous incompressible flows were found using RDTM.The results that we obtained are better than others, refer to the results in [4,8] in accuracy, convergence, and CPU time.
The structure of this paper is organized as follows: In Section 2, we begin with some basic definitions and the use of the RDTM on the KRLNS equations.Section 3 explains the manner we adopted to discuss the convergence of the solutions.In Section 4, we apply this method to solve four flow problems of different dimensions in order to show its ability and efficiency in finding new approximate solutions.Section 5 introduces conclusions of the present work.

Reduced Differential Transform Method
The RDTM is an iterative procedure for obtaining a Taylor series solution of differential equations.This method is similar to the differential transform method which was first introduced by Zhou [9].RDTM has been successfully used to many nonlinear problems [10][11][12][13][14][15][16][17][18][19] since it does not require any parameter, discretization, linearization, or small perturbations; thus it reduces the size of computational work and is easily applicable.
The main idea of this method depends on the representation the function of two variables (, ) as a product of single-variable function, i.e., (, ) = ()(), then the function (, ) can be represented as Definition .If function (, ) is analytic and differentiated continuously with respect to time  and space , then is called t-dimensional spectrum function of (, ), and is the transformed this function.
Definition .The reduced differential inverse transform of   () is defined as Then, the inverse transformation of the set of   () gives the −terms approximation solution as follows: and the exact solution is To show some basic properties of ( + 1) dimensional RDTM [18], we have to consider  = ( 1 ,  2 , . . .,   ) to be a vector of  variables, and function (, ) is analytic and continuously differentiable with respect to time  and space in the domain of interest, then the fundamental mathematical operations performed by RDTM are readily obtained and listed in Table 1.

Analysis of Convergence
The convergence of the approximate analytical solutions that are resulted from the application of RDTM to INS equations is discussed by relying on the approach followed in [20,21].
Let us consider the Hilbert space  =  2 ((, ) 3 × [0, ]) define by 2 (, )   < ∞, (16) and the norm We consider the INS equation in the following form: which is equivalent to the following form: where L =   is the linear partial derivative with respect to t, N is a nonlinear operator, R is a linear operator, and F is a general nonlinear operator involving both linear and nonlinear terms.According to RDTM and the solutions where It is noted that the solutions by RDTM are equivalent to determining the sequence . . .
The sufficient condition for convergence of the series solution {S  } ∞ 0 is presented in the following theorems Theorem 3. e series solution {S  = (  ,   ,   )} ∞ 0 converges whenever there is  such that 0 <  < 1,  =  1 + since ‖B 0 ‖ < ∞ and 0 <  < 1, we get lim ,→∞ ‖S  − S  ‖ = 0; thus, we conclude that {S  } ∞ 0 is a Cauchy sequence in the Hilbert space  3 , then the series solution {S  } ∞ 0 converges to some {S} ∈  3 .Theorem 4. Let F = (F 1 , F 2 , F 3 ) be a nonlinear operator satisfies Lipschitz condition from a Hilbert space  3 into  3 and u(, ) be the exact solution of INS equations.If the series solution {S  } ∞ 0 converges, then it is converged to u(, ).

Test Problems
In this section, the KRDTM is applied to find approximate analytical solutions of four unsteady viscous incompressible flow problems, two of these problems have exact solutions and the others do not have the exact solutions.We applied KRDTM for each problem to get some approximate analytical solutions.Then the convergence of these solutions has been discussed theoretically and numerically.Finally, the results have been reviewed through some figures, which represent the velocity components and the vorticity functions, which satisfy and explain the time development with the enstrophy, which is defined as where  is volume for three-dimension flow problems.Our results are computed by using various value of Reynolds numbers at some time levels.All our calculations are run by Maple 18 software.First problem (P1) is two-dimensional Taylor decaying vortices flow [4,7,8], which describes an initially periodical vortex structure convected by the flow field and exponentially decaying due to the viscous decaying.The exact solution of this problem that achieves (1) and ( 2) is when we used KRDTM for solving two-dimensional ( 1) and ( 2) equations, where  = (, ), u = (, V), U  = (  ,   ), and   := ; we get such that where  = 0, 1, 2, 3, . ..,  0 (, ) = (, , 0),  0 (, ) = V(, , 0), and  0 (, ) = (, , 0); the solutions are produced as follows: where such that and To prove that the condition of the convergence of these solutions verified on the domain [0, 2] and if  = 1000 then if  = 2 and  = 100 then and if  = 1000 then The errors measurements  1 ,  2 , and  ∞ -norm resulting from the application of KRDTM and the implicit central compact method (ICCM) in [8] for the computed  velocity component with CPU time for various grids at time level  = 0.5,  = 0.001, and  = 100 are tabulated in Table 2.In Table 3,  2 -norm errors for  are calculated at time level  = 5 and  = 0.001 for various Reynolds numbers.The contours of the vorticity and pressure are explained in Figure 1 at  = 2,  = 0.01, and  = 100.The comparisons of the computed  and V velocity components with the exact solution along the vertical and horizontal center lines at time levels  = 2,  = 0.01, and  = 100 are shown in Figure 2.
In Table 2, it can be noticed that the accuracy of new solutions and the size of the calculated errors of KRDTM are not often affected by the grid size which has been used in comparison with numerical results.Moreover, the results of KRDTM are better than ICCM [8].Also, from this table, it is clearly shown that the KRDTM results in less computation time (CPU) than ICCM [8].In Table 3, the same facts have been shown with various Reynolds numbers and grid spacing for  2 -norm at  = 5, where in some cases the CPU time reached zero.The results are given in Figure 1 show that the profiles of vorticity and pressure as contour plot are equivalent and identical with other results in [4,7,8].Moreover Figure 2 compares between the exact and new approximate analytical solutions, and the identical is confirmation of the efficiency of KRDTM in solving INS equations with good convergence for different time.
Second problem (P2) is Kovasznay flow [4,7,8], which is the laminar flow of viscous fluid behind a two-dimensional grid, with -axis normal to the grid and the velocity field is assumed to be such that  := U +  and V := V, where (, , ) and V(, , ) are the components of velocity; U is the average velocity in the -direction.Thus, the twodimensional INS equations with a periodicity in one direction may represent the wake of a two-dimensional grid the same as (1) with replacing the convective terms by ((U+, V)⋅∇)(, V), where U refers to one in this test.When we solved twodimensional equations ( 1) and ( 2) by using KRDTM for this test problem, we get the same equations ( 31 The exact solution of the steady state of this problem [7] considers the initial conditions in this test  where  = /2 − √ () 2 + 16 2 /2,  0 is a reference pressure (an arbitrary constant), and  0 = 0 in the test.The solutions produce similar solutions in (35), (36), and (37) with and . . .
In Tables 4 and 5, the  2 -norm error for the computed velocity component at  = 0.01 for some values of Reynolds numbers is compared with the numerical results of the upwind compact finite difference method (UCFDM) in [4].We noticed that the accuracy of the results obtained from KRDTM is higher and better than the results of UCFDM for different values of Reynolds numbers.It is clear that the maximum CPU time for all cases is not more than 30.0s, and iterations number (3) of KRDTM is less than iterations number of UCFDM (number of iterations ⩽ 3000).This shows that KRDTM is faster convergence and more accurate than UCFDM.The influence of the Reynolds number value on the computed vorticity and stream function (, , ) which is satisfied    (, , ) =  (, , ) ,    (, , ) = −V (, , ) is explained in Figure 3, so that the pairs of bound eddies produced behind the single elements of the grids and at large distance downstream.However, the streamlines are parallel and equidistant as shown by the short lines on the right side of the figure for all values of Reynolds numbers.We also note that when the value of Reynolds number increases, the whole flow pattern is expanded uniformly in the direction of main flow.Moreover, it can be observed that the rate of change of the flow is very great, and the length of vortices increases towards the downstream flow with the increase in the Reynolds number.
Third problem (P3) is three-dimensional Taylor decaying vortices flow, whose initial conditions [22][23][24][25][26] are given by with periodic boundary conditions in all directions, where  0 is a reference pressure (an arbitrary constant), U is characteristic velocity,  0 is the density, and  is the inverse of the wave number of the minimum frequencies (the largest length scale of flow).We used in this test  0 = 0, U = 1,  0 = 1, and  = 1 [23] and applied the KRDTM for solving three-dimensional equations ( 1) and ( 2), where  = (, , ), u = (, V, ), and U  = (  ,   ,   ), and we get where = 0, 1, 2, 3, . ..,  0 (, , ) = (, , , 0),  0 (, , ) = V(, , , 0),  0 (, , ) = (, , , 0), and  0 (, , ) = (, , , 0).Then the exact solution is obtained as follows: where such that and The flow is computed with in a periodic square box defined as − < , ,  < .The condition of the convergence of these solutions is verified by applying Definition 5  4 at  = 20, 40, 100, 500 for  = 0.1, 0.01.In Figure 5, we explained the change in the contours of the z-component of the vorticity and the velocities with time on the surface  = 0 at  = 100 and  = 0.05.Fourth problem (P4) is one type of three-dimension Beltrami flow [25,27,28], which yield a family of velocity and pressure fields depending on the selection of  and .In this test, we selected  = /4 and  = /2.This problem has the exact solution satisfying (1) and ( 2 if  = 5 then if  = 1600 and  = 2 then and if  = 5 then The  2 -norm error for the  velocity component with CPU time is calculated in Table 8 to study the accuracy of these approximate solutions; the results show an excellent accuracy of our method for all values of Reynolds number at  = 0.5, 2, 5 and  = 0.01, with good implementation period ranging between 2.09, 1080.The computed enstrophy is compared with their exact values in the same period of time in Figure 6 at  = 100, 1600 for two Mach numbers.In Figure 7, we explained the −component of the computed vorticity on surface  = 0 at  = 100 and  = 5 in two domains [−1, 1] 2 and [−5, 5] 2 .Through these figures, we could notice the relationship between the accuracy of these approximate solutions and Mach numbers which is with decreasing Mach number.

Conclusions
In this paper, the simulations of two-and three-dimensional unsteady viscous incompressible flow problems are presented by using the kinetically reduced local Navier-Stokes equations with the reduced differential transform method.New approximate analytical solutions obtained by KRDTM are tested in terms of accuracy and convergence.The results show that the new solutions have good accuracy and convergence, especially with high Reynolds numbers and low Mach numbers.The comparison explained that the computational time of these solutions is faster than that of other numerical ), (32), and (33) with   =   (, )  +  ∑ =0   (, )  − (, )  ,   =  ∑ =0   (, )  − (, )  ,   =   (, )  +  ∑ =0   (, )  − (, )  ,   =  ∑ =0   (, )  − (, )  .
The streamlines

Table 6 :
The  2 -norm errors for  and  of P3 at  = 100 and  = 0.005.

Table 7 :
The  2 -norm errors for  and  of P3 at  = 100 and  = 0.01.