scholarly journals A CFD Tutorial in Julia: Introduction to Laminar Boundary-Layer Theory

Fluids ◽  
2021 ◽  
Vol 6 (6) ◽  
pp. 207
Author(s):  
Furkan Oz ◽  
Kursat Kara
Keyword(s):  
Boundary Layer ◽  
Programming Languages ◽  
Programming Language ◽  
Laminar Boundary Layer ◽  
Aerodynamic Heating ◽  
Published Data ◽  
Viscous Effects ◽  
Flow Equations ◽  
Boundary Layer Equations ◽  
Flow Problems

Numerical simulations of laminar boundary-layer equations are used to investigate the origins of skin-friction drag, flow separation, and aerodynamic heating concepts in advanced undergraduate- and graduate-level fluid dynamics/aerodynamics courses. A boundary-layer is a thin layer of fluid near a solid surface, and viscous effects dominate it. Students must understand the modeling of flow physics and implement numerical methods to conduct successful simulations. Writing computer codes to solve equations numerically is a critical part of the simulation process. Julia is a new programming language that is designed to combine performance and productivity. It is dynamic and fast. However, it is crucial to understand the capabilities of a new programming language before attempting to use it in a new project. In this paper, fundamental flow problems such as Blasius, Hiemenz, Homann, and Falkner-Skan flow equations are derived from scratch and numerically solved using the Julia language. We used the finite difference scheme to discretize the governing equations, employed the Thomas algorithm to solve the resulting linear system, and compared the results with the published data. In addition, we released the Julia codes in GitHub to shorten the learning curve for new users and discussed the advantages of Julia over other programming languages. We found that the Julia language has significant advantages in productivity over other coding languages. Interested readers may access the Julia codes on our GitHub page.

scholarly journals A CFD Tutorial in Julia: Introduction to Compressible Laminar Boundary-Layer Flows

Fluids ◽  
2021 ◽  
Vol 6 (11) ◽  
pp. 400
Author(s):  
Furkan Oz ◽  
Kursat Kara
Keyword(s):  
Boundary Layer ◽  
Programming Language ◽  
Laminar Boundary Layer ◽  
Stokes Equations ◽  
Fluid Layer ◽  
Navier Stokes ◽  
Aerodynamic Heating ◽  
Boundary Layer Flows ◽  
Viscous Effects ◽  
Navier Stokes Equations

A boundary-layer is a thin fluid layer near a solid surface, and viscous effects dominate it. The laminar boundary-layer calculations appear in many aerodynamics problems, including skin friction drag, flow separation, and aerodynamic heating. A student must understand the flow physics and the numerical implementation to conduct successful simulations in advanced undergraduate- and graduate-level fluid dynamics/aerodynamics courses. Numerical simulations require writing computer codes. Therefore, choosing a fast and user-friendly programming language is essential to reduce code development and simulation times. Julia is a new programming language that combines performance and productivity. The present study derived the compressible Blasius equations from Navier–Stokes equations and numerically solved the resulting equations using the Julia programming language. The fourth-order Runge–Kutta method is used for the numerical discretization, and Newton’s iteration method is employed to calculate the missing boundary condition. In addition, Burgers’, heat, and compressible Blasius equations are solved both in Julia and MATLAB. The runtime comparison showed that Julia with for loops is 2.5 to 120 times faster than MATLAB. We also released the Julia codes on our GitHub page to shorten the learning curve for interested readers.

Optimal Performance and Geometry of Supersonic Ejector

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
N. I. I. Hewedy ◽  
Mofreh H. Hamed ◽  
F. Sh. Abou-Taleb ◽  
Tarek A. Ghonim
Keyword(s):  
Boundary Layer ◽  
Pressure Ratio ◽  
Flow Region ◽  
Operating Conditions ◽  
Maximum Efficiency ◽  
Published Data ◽  
Flow Equations ◽  
Boundary Layer Equations ◽  
Wall Boundary Layer ◽  
Potential Flows

The optimum geometries of the ejectors, which give maximum efficiency, are numerically predicted and experimentally measured. The numerical investigation is based on flow equations governing turbulent, compressible, two-dimensional, steady, time averaged, and boundary layer equations. These equations are iteratively solved using finite-difference method under the conditions of different flow regimes, which can be divided into several distinctive regions where the methods for estimating the mixing length are different for each flow region. The first region depicts the wall boundary layer, jet shear layer, and secondary and primary potential flows. The second one contains a single region of developing flow. A simple ejector with convergent-divergent primary nozzle was fabricated and experimentally tested. The present theoretical and experimental results are well compared with published data. The results obtained are used to correlate the optimum ejector geometry, pressure ratio, and ejector optimum efficiency as functions of the operation parameters and ejector area ratio. The resultant correlations help us to select the optimum ejector geometry and its corresponding maximum efficiency for particular operating conditions.

The Estimation of Laminar Skin Friction, including the Effects of Distributed Suction

1960 ◽  
Vol 11 (1) ◽  
pp. 1-21 ◽  
Author(s):  
N. Curle
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Laminar Boundary Layer ◽  
Simple Formula ◽  
Outer Part ◽  
Order Correction ◽  
Viscous Effects ◽  
Viscous Forces ◽  
Exact Theory ◽  
Total Head

SummaryStratford's analysis of the laminar boundary layer near separation uses two physical ideas. In the outer part of the boundary layer, where viscous effects are small, the development is given by the condition that the total head is constant along streamlines, apart from a second-order correction for viscosity. Near the wall, however, viscous forces must balance the pressure forces, and the profile adjusts itself accordingly. Quantitatively these ideas yield a simple formula for predicting separation, which has been found to be particularly accurate.In this paper it is indicated how the same approach may be used to yield the full distribution of skin friction along the wall. Further, the effects of suction may be incorporated into the method. Physically, suction affects the outer part of the boundary layer in that the streamlines are drawn towards the wall when suction is applied. At the wall, the balance between viscous and pressure forces is influenced by the momentum of the fluid which is sucked away. When these effects are accounted for quantitatively, the resulting formula for the skin friction is still very simple.Several examples are considered, and comparison is made with exact theory and with approximate results by other methods. It is indicated that the method has a useful range of validity.

Compressible Boundary Layer-Inviscid Flow Interactions in Entrance Region of Internal Flows

1967 ◽  
Vol 89 (4) ◽  
pp. 281-288 ◽  
Author(s):  
V. D. Blankenship ◽  
P. M. Chung
Keyword(s):  
Boundary Layer ◽  
Shock Tube ◽  
Inviscid Flow ◽  
Perturbation Parameter ◽  
Entrance Region ◽  
Compressible Boundary Layer ◽  
Internal Flows ◽  
Flow Equations ◽  
Boundary Layer Equations ◽  
Interaction Problem

The coupling between the inviscid flow and the compressible boundary layer in the developing entrance region for internal flows is analyzed by solving the particular inviscid flow-boundary layer interaction problem. The interaction problem is solved by postulating certain series forms of solutions for the inviscid region and the boundary layer. The boundary-layer equations and inviscid-flow equations are perturbed to third order and each generated equation is solved numerically. In order to preserve the universality of each of the perturbed boundary-layer equations, the perturbation parameter is described by an integral equation which is also solved in series form. The final results describing the interaction problem are then constructed for any given conditions by forming the three series to a consistent order of magnitude. This technique of coordinate perturbation is generalized to show how it may be applied to the entrance regions of pipe flows, including mass injection or suction, and also to the laminar boundary layers in shock tube flows. It demonstrates analytically the manner in which the boundary layer and inviscid flow interact and create a streamwise pressure gradient. In particular, the interaction problem which occurs in shock tube flows is solved in detail by the use of this generalized method, as an example.

The Critical Height of Distributed Roughness to Cause Boundary Layer Transition

1959 ◽  
Vol 63 (588) ◽  
pp. 722-722
Author(s):  
R. L. Dommett
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Laminar Boundary Layer ◽  
Boundary Layer Transition ◽  
Critical Height ◽  
Layer Transition ◽  
Local Conditions ◽  
Additional Advantage ◽  
Boundary Layer Equations ◽  
General Method

It has been found that there is a critical height for “sandpaper” type roughness below which no measurable disturbances are introduced into a laminar boundary layer and above which transition is initiated at the roughness. Braslow and Knox have proposed a method of predicting this height, for flow over a flat plate or a cone, using exact solutions of the laminar boundary layer equations combined with a correlation of experimental results in terms of a Reynolds number based on roughness height, k, and local conditions at the top of the elements. A simpler, yet more general, method can be constructed by taking additional advantage of the linearity of the velocity profile near the wall in a laminar boundary layer.

On the propagation of disturbances in a laminar boundary layer. I

1973 ◽  
Vol 73 (3) ◽  
pp. 493-503 ◽  
Author(s):  
S. N. Brown ◽  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Asymptotic Formula ◽  
Boundary Conditions ◽  
Laminar Boundary Layer ◽  
Numerical Results ◽  
Boundary Layer Equations ◽  
Displacement Thickness ◽  
Layer I

An analysis is made of the response of a laminar boundary layer to a perturbation, either in the mainstream, or of the boundary conditions at the wall. The disturbance propagates with the mainstream velocity, and the manner in which it decays at large distances downstream is determined by eigensolutions of the boundary-layer equations. The elucidation of the structure of these eigensolutions requires division of the boundary layer into three regions. Comparison of the asymptotic formula obtained for the displacement thickness is made with the numerical results of Ackerberg and Phillips (1).

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