A numerical method for integrating the unsteady boundary-layer equations when there are regions of backflow

1973 ◽  
Vol 58 (3) ◽  
pp. 561-579 ◽  
Author(s):  
J. H. Phillips ◽  
R. C. Ackerberg
Keyword(s):  
Boundary Layer ◽  
Numerical Method ◽  
Unique Feature ◽  
Iterative Technique ◽  
Unsteady Boundary Layer ◽  
Boundary Layer Equations ◽  
Computation Step ◽  
Finite Difference Equations ◽  
Mesh Spacing ◽  
Time Variables

A numerical method for integrating the unsteady twodimensional boundarylayer equations using a second-order-accurate implicit method, which allows for arbitrary mesh spacing in the space and time variables, is developed. A unique feature of the method is the use of an asymptotic solution valid at the downstream end of the integration mesh which permits backflow to be taken into account. Newton's iterative technique is used to solve the nonlinear finite-difference equations a t each computation step, using a rapid algorithm for solving the resulting linearized equations. The method is applied to a flow which is periodic in time and contains regions of backflow. The numerical computations are compared with known numerical and asymptotic solutions and the agreement is excellent.

An effective numerical method for solving viscous–inviscid interaction problems

2005 ◽  
Vol 363 (1830) ◽  
pp. 1157-1167 ◽  
Author(s):  
Marina A Kravtsova ◽  
Vladimir B Zametaev ◽  
Anatoly I Ruban
Keyword(s):  
Boundary Layer ◽  
Numerical Method ◽  
Corner Point ◽  
Boundary Layer Separation ◽  
Separated Flows ◽  
Governing Equations ◽  
Body Contour ◽  
Boundary Layer Equations ◽  
Finite Difference Equations ◽  
Dimensional Boundary

This paper presents a new numerical method to solve the equations of the asymptotic theory of separated flows. A number of measures was taken to ensure fast convergence of the iteration procedure, which is employed to treat the nonlinear terms in the governing equations. Firstly, we selected carefully the set of variables for which the nonlinear finite difference equations were formulated. Secondly, a Newton–Raphson strategy was applied to these equations. Thirdly, the calculations were facilitated by utilizing linear approximation of the boundary-layer equations when calculating the corresponding Jacobi matrix. The performance of the method is illustrated, using as an example, the problem of laminar two-dimensional boundary-layer separation in the flow of an incompressible fluid near a corner point of a rigid body contour. The solution of this problem is non-unique in a certain parameter range where two solution branches are possible.

scholarly journals Numerical Solutions of Unsteady Boundary Layer Flow with a Time-Space Fractional Constitutive Relationship

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1446
Author(s):  
Weidong Yang ◽  
Xuehui Chen ◽  
Yuan Meng ◽  
Xinru Zhang ◽  
Shiyun Mi
Keyword(s):  
Boundary Layer ◽  
Numerical Method ◽  
Boundary Layer Flow ◽  
Numerical Solutions ◽  
Constitutive Relationship ◽  
Elastic Behavior ◽  
Unsteady Boundary Layer ◽  
Time Space ◽  
Layer Flow ◽  
Fractional Parameter

In this paper, we develop a new time-space fractional constitution relation to study the unsteady boundary layer flow over a stretching sheet. For the convenience of calculation, the boundary layer flow is simulated as a symmetrical rectangular area. The implicit difference method combined with an L1-algorithm and shift Grünwald scheme is used to obtain the numerical solutions of the fractional governing equation. The validity and solvability of the present numerical method are analyzed systematically. The numerical results show that the thickness of the velocity boundary layer increases with an increase in the space fractional parameter γ. For a different stress fractional parameter α, the viscoelastic fluid will exhibit viscous or elastic behavior, respectively. Furthermore, the numerical method in this study is validated and can be extended to other time-space fractional boundary layer models.

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