On the numerical computation of the optimum profile in Stokes flow

1974 ◽  
Vol 65 (3) ◽  
pp. 513-515 ◽  
Author(s):  
J.-M. Bourot
Keyword(s):  
Numerical Computation ◽  
Stokes Flow ◽  
Stokes Equations ◽  
The Body ◽  
Slip Condition ◽  
Quadratic Minimization ◽  
Optimum Profile ◽  
Theoretical Results

The body of given volume with the smallest drag in Stokes flow is obtained by making use of theoretical results due to Pironneau. A suitable family of solutions of the Stokes equations is used and the no-slip condition is expressed numerically by a technique of quadratic minimization. We find that the drag on this optimal body is 0·95425 times the drag on the sphere of equal volume.

scholarly journals Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows

1975 ◽  
Vol 67 (4) ◽  
pp. 787-815 ◽  
Author(s):  
Allen T. Chwang ◽  
T. Yao-Tsu Wu
Keyword(s):  
Exact Solutions ◽  
Stokes Flow ◽  
Fundamental Solutions ◽  
The Body ◽  
Circular Cylinders ◽  
Geometrical Parameters ◽  
Flow Problems ◽  
Singularity Method ◽  
Wide Range ◽  
Body Shapes

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.

On the Flow Fields of Inertial Impactors

1974 ◽  
Vol 96 (4) ◽  
pp. 394-400 ◽  
Author(s):  
V. A. Marple ◽  
B. Y. H. Liu ◽  
K. T. Whitby
Keyword(s):  
Flow Field ◽  
Stokes Equations ◽  
Relaxation Method ◽  
Water Model ◽  
Navier Stokes ◽  
Visualization Technique ◽  
Navier Stokes Equations ◽  
Theoretical Results ◽  
Plate Distance ◽  
Good Agreement

The flow field in an inertial impactor was studied experimentally with a water model by means of a flow visualization technique. The influence of such parameters as Reynolds number and jet-to-plate distance on the flow field was determined. The Navier-Stokes equations describing the laminar flow field in the impactor were solved numerically by means of a finite difference relaxation method. The theoretical results were found to be in good agreement with the empirical observations made with the water model.

Lift and power requirements of hovering flight inDrosophila virilis

2002 ◽  
Vol 205 (16) ◽  
pp. 2413-2427 ◽  
Author(s):  
Mao Sun ◽  
Jian Tang
Keyword(s):  
Flight Control ◽  
Stokes Equations ◽  
Fruit Fly ◽  
The Body ◽  
Drosophila Virilis ◽  
Specific Power ◽  
Aerodynamic Forces ◽  
Hovering Flight ◽  
Power Expenditure ◽  
The Mean

SUMMARYThe lift and power requirements for hovering flight in Drosophila virilis were studied using the method of computational fluid dynamics. The Navier-Stokes equations were solved numerically. The solution provided the flow velocity and pressure fields, from which the unsteady aerodynamic forces and moments were obtained. The inertial torques due to the acceleration of the wing mass were computed analytically. On the basis of the aerodynamic forces and moments and the inertial torques, the lift and power requirements for hovering flight were obtained.For the fruit fly Drosophila virilis in hovering flight (with symmetrical rotation), a midstroke angle of attack of approximately 37°was needed for the mean lift to balance the insect weight, which agreed with observations. The mean drag on the wings over an up- or downstroke was approximately 1.27 times the mean lift or insect weight (i.e. the wings of this tiny insect must overcome a drag that is approximately 27 % larger than its weight to produce a lift equal to its weight). The body-mass-specific power was 28.7 W kg-1, the muscle-mass-specific power was 95.7 W kg-1 and the muscle efficiency was 17 %.With advanced rotation, larger lift was produced than with symmetrical rotation, but it was more energy-demanding, i.e. the power required per unit lift was much larger. With delayed rotation, much less lift was produced than with symmetrical rotation at almost the same power expenditure; again, the power required per unit lift was much larger. On the basis of the calculated results for power expenditure, symmetrical rotation should be used for balanced, long-duration flight and advanced rotation and delayed rotation should be used for flight control and manoeuvring. This agrees with observations.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

scholarly journals The inf-sup stability of the lowest order Taylor–Hood pair on affine anisotropic meshes

2019 ◽  
Vol 40 (4) ◽  
pp. 2377-2398
Author(s):  
Gabriel R Barrenechea ◽  
Andreas Wachtel
Keyword(s):  
Finite Element ◽  
Fluid Mechanics ◽  
Incompressible Fluid ◽  
Stokes Equations ◽  
Sufficient Conditions ◽  
Navier Stokes ◽  
Element Solution ◽  
Navier Stokes Equations ◽  
Anisotropic Meshes ◽  
Theoretical Results

Abstract Uniform inf-sup conditions are of fundamental importance for the finite element solution of problems in incompressible fluid mechanics, such as the Stokes and Navier–Stokes equations. In this work we prove a uniform inf-sup condition for the lowest-order Taylor–Hood pairs $\mathbb{Q}_2\times \mathbb{Q}_1$ and $\mathbb{P}_2\times \mathbb{P}_1$ on a family of affine anisotropic meshes. These meshes may contain refined edge and corner patches. We identify necessary hypotheses for edge patches to allow uniform stability and sufficient conditions for corner patches. For the proof, we generalize Verfürth’s trick and recent results by some of the authors. Numerical evidence confirms the theoretical results.

Flow induced by jets and plumes

1981 ◽  
Vol 108 ◽  
pp. 55-65 ◽  
Author(s):  
W. Schneider
Keyword(s):  
Stokes Equations ◽  
Outer Region ◽  
Reynolds Numbers ◽  
Slip Condition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Laminar Jet ◽  
Order Of Magnitude ◽  
Valid Solution ◽  
Solid Walls

The order of magnitude of the flow velocity due to the entrainment into an axisymmetric, laminar or turbulent jet and an axisymmetric laminar plume, respectively, indicates that viscosity and non-slip of the fluid at solid walls are essential effects even for large Reynolds numbers of the jet or plume. An exact similarity solution of the Navier-Stokes equations is determined such that both the non-slip condition at circular-conical walls (including a plane wall) and the entrainment condition at the jet (or plume) axis are satisfied. A uniformly valid solution for large Reynolds numbers, describing the flow in the laminar jet region as well as in the outer region, is also given. Comparisons show that neither potential flow theory (Taylor 1958) nor viscous flow theories that disregard the non-slip condition (Squire 1952; Morgan 1956) provide correct results if the flow is bounded by solid walls.

Studies on Natural Convection Induced Flow and Thermal Behavior Inside Electronic Equipment Cabinet Model

2001 ◽  
Author(s):  
Masaru Ishizuka ◽  
Guoyi Peng ◽  
Shinji Hayama
Keyword(s):  
Natural Convection ◽  
Thermal Behavior ◽  
Body Surface ◽  
Stokes Equations ◽  
The Body ◽  
Electronic Equipment ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Navier Stokes Equations ◽  
Induced Flow

Abstract In the present work, an important basic flow phenomena, the natural convection induced flow, is studied numerically. Three-dimensional Navier-Stokes equations along with the temperature equation are solved on the basis of finite difference method. Generalized coordinate system is used so that sufficient grid resolution could be achieved in the body surface boundary layer region. Differential terms with respect to time are approximated by forward differences, diffusions terms are approximated by the implicit Euler form, convection terms in the Navier-Stokes equations are approximated by the third order upwind difference scheme. The heat flux at the body surface of heater is specified. The results of calculation showed a satisfactory agreement with the measured data and led to a good understanding of the overall flow and thermal behavior inside electronic equipment cabinet model which is very difficult, if not impossible, to gather by experiment.

scholarly journals Full-Span Flying Wing Wind Tunnel Test: A Body Freedom Flutter Study

Fluids ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 34
Author(s):  
Pengtao Shi ◽  
Jihai Liu ◽  
Yingsong Gu ◽  
Zhichun Yang ◽  
Pier Marzocca
Keyword(s):  
Wind Tunnel ◽  
Theoretical Analysis ◽  
Mass Balance ◽  
Degrees Of Freedom ◽  
Wind Tunnel Test ◽  
Suspension System ◽  
The Body ◽  
Structural Dynamic ◽  
Flutter Speed ◽  
Theoretical Results

Aiming at the experimental test of the body freedom flutter for modern high aspect ratio flexible flying wing, this paper conducts a body freedom flutter wind tunnel test on a full-span flying wing flutter model. The research content is summarized as follows: (1) The full-span finite element model and aeroelastic model of an unmanned aerial vehicle for body freedom flutter wind tunnel test are established, and the structural dynamics and flutter characteristics of this vehicle are obtained through theoretical analysis. (2) Based on the preliminary theoretical analysis results, the design and manufacturing of this vehicle are completed, and the structural dynamic characteristics of the vehicle are identified through ground vibration test. Finally, the theoretical analysis model is updated and the corresponding flutter characteristics are obtained. (3) A novel quasi-free flying suspension system capable of releasing pitch, plunge and yaw degrees of freedom is designed and implemented in the wind tunnel flutter test. The influence of the nose mass balance on the flutter results is explored. The study shows that: (1) The test vehicle can exhibit body freedom flutter at low airspeeds, and the obtained flutter speed and damping characteristics are favorable for conducting the body freedom flutter wind tunnel test. (2) The designed suspension system can effectively release the degrees of freedom of pitch, plunge, and yaw. The flutter speed measured in the wind tunnel test is 9.72 m/s, and the flutter frequency is 2.18 Hz, which agree well with the theoretical results (with flutter speed of 9.49 m/s and flutter frequency of 2.03 Hz). (3) With the increasing of the mass balance at the nose, critical speed of body freedom flutter rises up and the flutter frequency gradually decreases, which also agree well with corresponding theoretical results.

scholarly journals EVALUATION OF THE APPLICABILITY OF A TURBULENT WAKE INLET BOUNDARY CONDITION

2017 ◽  
Vol 16 (2) ◽  
pp. 78
Author(s):  
P. A. Soliman ◽  
A. V. de Paula ◽  
A. P. Petry ◽  
S. V. Möller
Keyword(s):  
Stokes Equations ◽  
Transport Model ◽  
Characteristic Length ◽  
Computational Cost ◽  
The Body ◽  
Navier Stokes ◽  
Computational Time ◽  
Flow Forming ◽  
Shear Stress Transport Model ◽  
Grid Convergence Index

With the objective of reducing the computational cost of the iterative processes of aerodynamic components design, tests were carried out to study under what conditions, and with what difference, only part of the calculation domain can be solved using as input information obtained from complete simulations already solved. An experimental study of an airfoil exposed to the wake interference of an upstream airfoil at a Reynolds number of 150,000 was used to verify the solutions of the Reynolds-Averaged Navier-Stokes equations solved applying the k-ω Shear Stress Transport model for turbulence closure. A Grid Convergence Index study was performed to verify if the solution of the equations for the adopted discretization leads to results within the asymptotic range. With the physical coherence of the numerical methodology verified, comparisons between the simulations with the domain comprising the two airfoils and the domain comprising only the downstream airfoil were performed. Computational time reductions in the order of 40% are observed. The differences in the aerodynamic coefficients for the two types of simulation are presented as a function of distances non-dimensionalized by the characteristic length of the body that disturbs the flow forming the wake, showing that the difference between the two methods was inversely proportional to the distance between the two bodies. Behavior that was maintained until a point where the simulation diverges, equivalent to 25% of the characteristic length of the body that generates the wake.

Studies on natural convection-induced flow and thermal behaviour inside electronic equipment cabinet model

2003 ◽  
Vol 217 (3) ◽  
pp. 323-328 ◽  
Author(s):  
M Ishizuka ◽  
Y Kitamura
Keyword(s):  
Natural Convection ◽  
Thermal Behaviour ◽  
Stokes Equations ◽  
Three Dimensional ◽  
The Body ◽  
Electronic Equipment ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Surface Boundary Layer ◽  
Induced Flow

In the present work, an important basic flow phenomenon, natural convection-induced flow, is studied numerically. Three-dimensional Navier-Stokes equations along with the energy equation are solved based on the finite difference method. A generalized coordinate system is used so that sufficient grid resolution could be achieved in the body surface boundary layer region. The results of calculation showed a satisfactory agreement with the measured data and led to a good understanding of the overall flow and thermal behaviour inside an electronic equipment cabinet model, which is very difficult, if not impossible, to gather by experiment.

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