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.

Compressibility Effects on the Extended Crocco Relation and the Thermal Recovery Factor in Laminar Boundary Layer Flow

2004 ◽  
Vol 126 (1) ◽  
pp. 32-41 ◽  
Author(s):  
B. W. van Oudheusden
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Pressure Gradient ◽  
Boundary Layer Flow ◽  
Numerical Solutions ◽  
Thermal Recovery ◽  
Recovery Factor ◽  
Perturbation Approach ◽  
Layer Flow ◽  
Compressibility Effects

The relation between velocity and enthalpy in steady boundary layer flow is known as the Crocco relation. It describes that for an adiabatic wall the total enthalpy remains constant throughout the boundary layer, when the Prandtl number (Pr) is one, irrespective of pressure gradient and compressibility. A generalization of the Crocco relation for Pr near one is obtained from a perturbation approach. In the case of constant-property flow an analytic expression is found, representing a first-order extension of the standard Crocco relation and confirming the asymptotic validity of the square-root dependence of the recovery factor on Prandtl number. The particular subject of the present study is the effect of compressibility on the extended Crocco relation and, hence, on the thermal recovery in laminar flows. A perturbation analysis for constant Pr reveals two additional mechanisms of compressibility effects in the extended Crocco relation, which are related to the viscosity law and to the pressure gradient. Numerical solutions for (quasi-)self-similar as well as non-similar boundary layers are presented to evaluate these effects quantitatively.

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