Recent Numerical Advances in Fluid Mechanics, Volume II

A special issue of Fluids (ISSN 2311-5521). This special issue belongs to the section "Mathematical and Computational Fluid Mechanics".

Deadline for manuscript submissions: closed (31 January 2021) | Viewed by 44203

Special Issue Editor

Mechanical and Aerospace Engineering, Oklahoma State University, 201 General Academic Building, Stillwater, OK 74078-5016, USA
Interests: fluid mechanics; complex systems; pattern formation; partial differential equations; non-Newtonian fluids; fluid–structure interaction; hydrodynamic stability; non-equilibrium thermodynamics; vortex induced oscillations; rheology; pathological flows; network analysis; philosophy of science; sustainability and science and creativity in mathematics and science
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Special Issue Information

Dear Colleagues,

In recent decades, the field of computational fluid dynamics has made significant advances in enabling advanced computing architectures to understand many phenomena in biological, geophysical, and engineering fluid flows. Almost all research areas in fluids use numerical methods at various complexities: from molecular to continuum descriptions; from laminar to turbulent regimes; from low-speed to hypersonic, from stencil-based computations to meshless approaches; from local basis functions to global expansions, as well as from 1st-order approximation to high order and spectral accuracy. Many successful efforts have been put forth in dynamic adaptation strategies, e.g., adaptive mesh refinement and multiresolution representation approaches. Furthermore, with recent advances in artificial intelligence and heterogeneous computing, broader fluids community has gained momentum to revisit and investigate such practices. In this Special Issue, we aim to bring together researchers to provide a state of the art overview of the current investigations and topics on computational fluid dynamics.

Dr. Omer San
Guest Editor

Manuscript Submission Information

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Keywords

  • computational fluid dynamics
  • finite difference, finite element, finite volume methods
  • spectral difference, spectral element, spectral volume methods
  • discontinuous galerkin methods
  • lattice boltzmann method
  • smoothed particle hydrodynamics
  • immersed boundary methods
  • multigrid methods
  • wavelets
  • adaptive mesh refinement
  • meshless methods
  • shock capturing
  • hybrid dissipative and non-dissipative approaches
  • high performance scientific computing
  • heterogeneous computing
  • quantum computing
  • synchronous and asynchronous algorithms
  • numerical linear algebra
  • turbulence modeling
  • intelligent numerical methods
  • uncertainty quantification

Published Papers (15 papers)

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Research

16 pages, 3661 KiB  
Article
On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization—Part II: Viscous Flows
by Farzad Mohebbi, Ben Evans and Mathieu Sellier
Fluids 2021, 6(3), 106; https://doi.org/10.3390/fluids6030106 - 4 Mar 2021
Cited by 3 | Viewed by 2689
Abstract
This study presents an extension of a previous study (On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization) to viscous transonic flows. In this work, we showed that the same procedure to derive an explicit expression for an exact step length β [...] Read more.
This study presents an extension of a previous study (On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization) to viscous transonic flows. In this work, we showed that the same procedure to derive an explicit expression for an exact step length βexact in a gradient-based optimization method for inviscid transonic flows can be employed for viscous transonic flows. The extended numerical method was evaluated for the viscous flows over the transonic RAE 2822 airfoil at two common flow conditions in the transonic regime. To do so, the RAE 2822 airfoil was reconstructed by a Bezier curve of degree 16. The numerical solution of the transonic turbulent flow over the airfoil was performed using the solver ANSYS Fluent (using the Spalart–Allmaras turbulence model). Using the proposed step length, a gradient-based optimization method was employed to minimize the drag-to-lift ratio of the airfoil. The gradient of the objective function with respect to design variables was calculated by the finite-difference method. Efficiency and accuracy of the proposed method were investigated through two test cases. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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16 pages, 966 KiB  
Article
Fast-Projection Methods for the Incompressible Navier–Stokes Equations
by Carlo De Michele, Francesco Capuano and Gennaro Coppola
Fluids 2020, 5(4), 222; https://doi.org/10.3390/fluids5040222 - 27 Nov 2020
Cited by 5 | Viewed by 2611
Abstract
An analysis of existing and newly derived fast-projection methods for the numerical integration of incompressible Navier–Stokes equations is proposed. Fast-projection methods are based on the explicit time integration of the semi-discretized Navier–Stokes equations with a Runge–Kutta (RK) method, in which only one Pressure [...] Read more.
An analysis of existing and newly derived fast-projection methods for the numerical integration of incompressible Navier–Stokes equations is proposed. Fast-projection methods are based on the explicit time integration of the semi-discretized Navier–Stokes equations with a Runge–Kutta (RK) method, in which only one Pressure Poisson Equation is solved at each time step. The methods are based on a class of interpolation formulas for the pseudo-pressure computed inside the stages of the RK procedure to enforce the divergence-free constraint on the velocity field. The procedure is independent of the particular multi-stage method, and numerical tests are performed on some of the most commonly employed RK schemes. The proposed methodology includes, as special cases, some fast-projection schemes already presented in the literature. An order-of-accuracy analysis of the family of interpolations here presented reveals that the method generally has second-order accuracy, though it is able to attain third-order accuracy only for specific interpolation schemes. Applications to wall-bounded 2D (driven cavity) and 3D (turbulent channel flow) cases are presented to assess the performances of the schemes in more realistic configurations. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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25 pages, 11218 KiB  
Article
Analysis of Cold Air Recirculation in the Evaporators of Large-Scale Air-Source Heat Pumps Using CFD Simulations
by Brice Rogié, Jonas Kjær Jensen, Svenn Ole Kjøller Hansen and Wiebke Brix Markussen
Fluids 2020, 5(4), 186; https://doi.org/10.3390/fluids5040186 - 21 Oct 2020
Cited by 5 | Viewed by 2984
Abstract
The present study investigates cold air recirculation in the evaporators of large-scale air-source heat pumps. A case study considered a 5 MW air-source heat pump producing heat for district heating. The heat pump comprises 20 horizontal evaporators, where each evaporator is equipped with [...] Read more.
The present study investigates cold air recirculation in the evaporators of large-scale air-source heat pumps. A case study considered a 5 MW air-source heat pump producing heat for district heating. The heat pump comprises 20 horizontal evaporators, where each evaporator is equipped with eight fans. The evaporators were implemented in a CFD model, where the influence of the outdoor wind direction on the recirculation was investigated. Firstly, the air recirculation was analysed with no surrounding obstacles. Secondly, the surrounding building and the real ground topology was included in the CFD model, to analyse their influence on the air recirculation. The results show that recirculation occurs for all wind directions, due to the turbulent behaviour of the flow around the evaporators. The results also show that the presence of a building intensifies the recirculation when it is placed directly upstream of the evaporators due to the presence of vortices in the wake of the building. On the other hand, a ground depression helps to reduce the recirculation by having additional energy dissipation due to the sudden change in the ground direction. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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14 pages, 2660 KiB  
Article
Flow Pressure Behavior Downstream of Ski Jumps
by Agostino Lauria, Giancarlo Alfonsi and Ali Tafarojnoruz
Fluids 2020, 5(4), 168; https://doi.org/10.3390/fluids5040168 - 29 Sep 2020
Cited by 15 | Viewed by 2202
Abstract
Ski jump spillways are frequently implemented to dissipate energy from high-speed flows. The general feature of this structure is to transform the spillway flow into a free jet up to a location where the impact of the jet creates a plunge pool, representing [...] Read more.
Ski jump spillways are frequently implemented to dissipate energy from high-speed flows. The general feature of this structure is to transform the spillway flow into a free jet up to a location where the impact of the jet creates a plunge pool, representing an area for potential erosion phenomena. In the present investigation, several tests with different ski jump bucket angles are executed numerically by means of the OpenFOAM® digital library, taking advantage of the Reynolds-averaged Navier–Stokes equations (RANS) approach. The results are compared to those obtained experimentally by other authors as related to the jet length and shape, obtaining physical insights into the jet characteristics. Particular attention is given to the maximum pressure head at the tailwater. Simple equations are proposed to predict the maximum dynamic pressure head acting on the tailwater, as dependent upon the Froude number, and the maximum pressure head on the bucket. Results of this study provide useful suggestions for the design of ski jump spillways in dam construction. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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31 pages, 8190 KiB  
Article
Parametric Study of Turbulent Couette Flow over Wavy Surfaces Using RANS Simulation: Effects of Aspect-Ratio, Wave-Slope and Reynolds Number
by Akshay Sherikar and Peter J. Disimile
Fluids 2020, 5(3), 138; https://doi.org/10.3390/fluids5030138 - 25 Aug 2020
Cited by 5 | Viewed by 3089
Abstract
A turbulent Couette flow over a wavy surface is subject to a detailed parametric study in which three parameters—Aspect Ratio, Wave Slope and Reynolds number—are independently varied over an order of magnitude to investigate their influence on the flow. Stdk−ε turbulence model [...] Read more.
A turbulent Couette flow over a wavy surface is subject to a detailed parametric study in which three parameters—Aspect Ratio, Wave Slope and Reynolds number—are independently varied over an order of magnitude to investigate their influence on the flow. Stdk−ε turbulence model with enhanced wall functions is used to simulate all cases in the study and the results are validated against experimental data as well as analytical theories pertaining to flow over wavy surfaces. Gross flow properties such as mean velocity profiles, mass flow rate, shear stress and pressure on the walls, as well as turbulent flow characteristics such as inner-wall coordinates, log-law fit, eddy viscosity profiles and turbulence kinetic energy across the domain, are presented and their corroboration with existing literature is discussed. The effect of the three parameters on the flow variables is investigated. It is observed that while all response flow variables scale monotonically with a progressive change in the parameters, there are certain flow characteristics that can be ascribed exclusively to one of the three parameters. The study also discusses the influence of the top plate, a much-needed discussion that seems to be absent in most literature pertaining to Couette flow in wavy channels. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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18 pages, 6392 KiB  
Article
Partially Averaged Navier-Stokes: A (k-ω)/(k-ε) Bridging Model
by Abdelkader Frendi and Christopher Harrison
Fluids 2020, 5(3), 129; https://doi.org/10.3390/fluids5030129 - 5 Aug 2020
Cited by 3 | Viewed by 3336
Abstract
A new Partially Averaged Navier-Stokes (PANS) bridging model is derived from existing (kω) and (kε) PANS formulations. The model behaves like the PANS (kω) model near rigid walls and like the [...] Read more.
A new Partially Averaged Navier-Stokes (PANS) bridging model is derived from existing (kω) and (kε) PANS formulations. The model behaves like the PANS (kω) model near rigid walls and like the PANS (kε) model away from walls. The new model is tested using well-known benchmark problems; a backward-facing step representing wall-bounded flows, and a circular cylinder representing free shear flows. Our results are compared to existing experimental data and previous simulation results using PANS (kω) and PANS (kε). The comparisons show our model to be superior at predicting velocity profiles in both flows. In addition, Reynolds stress predictions are also shown to improve. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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20 pages, 8901 KiB  
Article
Uncertainty Quantification Methodologies Applied to the Rotor Tip Clearance Effect in a Twin Scroll Radial Turbine
by Carlo Cravero and Andrea Ottonello
Fluids 2020, 5(3), 114; https://doi.org/10.3390/fluids5030114 - 17 Jul 2020
Cited by 4 | Viewed by 1726
Abstract
In the last three decades computer simulation tools have achieved wide spread use in the design and analysis of engineering devices. This has shortened the overall product design cycle (physical experiments may be impossible during early design stages) and it has also provided [...] Read more.
In the last three decades computer simulation tools have achieved wide spread use in the design and analysis of engineering devices. This has shortened the overall product design cycle (physical experiments may be impossible during early design stages) and it has also provided better understanding of the operating behavior of the systems under investigation. As a consequence numerical simulation have led to a reduction of physical prototyping and to lower costs for manufacturing production chains. Despite this success, it remains difficult to provide objective confidence levels in quantitative information derived from numerical predictions. The complexity arises from the amount of uncertainties related to the inputs of any computation attempting to represent a physical system. This paper focuses on geometrical sources of uncertainty in the field of CFD applied to twin scroll radial turbines. In particular it has been investigated the effect of uncertainties on tip clearance values at rotor blade leading edge and trailing edge on selected turbine performance parameters. The analysis shows the use of the Surrogate-based uncertainty quantification technique that has been setup by the authors in the Dakota® environment. The polynomial chaos expansion method has been applied to the same case. The comparison of the results coming from the different approaches and the discussion of the pros and cons related to each technique lead to interesting conclusions, which are proposed as guidelines for future UQ applications on the theme of CFD applied to radial turbines. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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21 pages, 3080 KiB  
Article
The Effects of Grid Accuracy on Flow Simulations: A Numerical Assessment
by Majid Allahyari, Vahid Esfahanian and Kianoosh Yousefi
Fluids 2020, 5(3), 110; https://doi.org/10.3390/fluids5030110 - 10 Jul 2020
Cited by 8 | Viewed by 4010
Abstract
High-quality, accurate grid generation is a critical challenge in the computational simulation of fluid flows around complex geometries. In particular, the accuracy of the grids is an effective factor in order to achieve a successful numerical simulation. In the current study, we present [...] Read more.
High-quality, accurate grid generation is a critical challenge in the computational simulation of fluid flows around complex geometries. In particular, the accuracy of the grids is an effective factor in order to achieve a successful numerical simulation. In the current study, we present a series of systematic numerical simulations for fluid flows around a NACA 0012 airfoil using different computational grid generation techniques, including the standard second-order, fourth-order compact, and Theodorsen transformation approaches, to assess the effects of grid accuracy on the flow solutions. The flow solvers are based on the second- and fourth-order schemes for spatial discretizations and Beam-Warming linearization method for time advancement. The obtained grids, as well as the metrics and the corresponding numerical flow solution for each grid generation technique, are compared and studied in detail. It is demonstrated that the quality and orthogonality of the grids is improved by using the fourth-order compact scheme. Moreover, the numerical assessment showed that the accuracy and the quality of the grids directly influence the numerical flow solutions. Finally, the higher-order accurate flow solvers are found to be more sensitive to the accuracy of the generated grid. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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15 pages, 280 KiB  
Article
A Note on Bi-Orthogonal Polynomials and Functions
by Clemente Cesarano
Fluids 2020, 5(3), 105; https://doi.org/10.3390/fluids5030105 - 30 Jun 2020
Cited by 10 | Viewed by 2279
Abstract
The theory of orthogonal polynomials is well established and detailed, covering a wide field of interesting results, as, in particular, for solving certain differential equations. On the other side the concepts and the related formalism of the theory of bi-orthogonal polynomials is less [...] Read more.
The theory of orthogonal polynomials is well established and detailed, covering a wide field of interesting results, as, in particular, for solving certain differential equations. On the other side the concepts and the related formalism of the theory of bi-orthogonal polynomials is less developed and much more limited. By starting from the orthogonality properties satisfied from the ordinary and generalized Hermite polynomials, it is possible to derive a further family (known in literature) of these kind of polynomials, which are bi-orthogonal with their adjoint. This aspect allows us to introduce functions recognized as bi-orthogonal and investigate generalizations of families of orthogonal polynomials Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
14 pages, 3881 KiB  
Communication
Development of a Passive Spore Sampler for Capture Enhancement of Airborne Crop Pathogens
by James L. Blackall, Jie Wang, Mostafa R. A. Nabawy, Mark K. Quinn and Bruce D. Grieve
Fluids 2020, 5(2), 97; https://doi.org/10.3390/fluids5020097 - 18 Jun 2020
Viewed by 4288
Abstract
Yellow rust spores currently blight commercial and domestic wheat production in areas of East Africa such as Ethiopia. Yellow rust is a hazard to crops which appears asymptomatic for a time, but inevitably causes significant losses in yield once symptoms of infection manifest [...] Read more.
Yellow rust spores currently blight commercial and domestic wheat production in areas of East Africa such as Ethiopia. Yellow rust is a hazard to crops which appears asymptomatic for a time, but inevitably causes significant losses in yield once symptoms of infection manifest themselves to the point where they can be readily observed by the naked eye. Regionally recurrent losses of up to 5% are common and reach as high as 25% in rare cases. Historically, spore sampling has been undertaken by large, cumbersome devices that require heavy power supplies and significant expertise to reliably operate. Moreover, tools for the design and development of such devices are currently limited. This paper, therefore, proposes design and testing processes to develop a spore sampling device that is compact, passive (requires no power to operate), and can better direct spores onto a biomimetic sensor platform enhancing the capture and detection of pathogens. This represents a novel design context for fluidic devices. Performance of the device has been simulated using Lagrangian particle tracking embedded into computational fluid dynamics (CFD) simulations, demonstrating significant improvements across a range of spore Stokes numbers. Experimental validation of numerical simulations was performed using wind tunnel testing and practical performance such as weathervaning was demonstrated. Results show that that the developed sampler is capable of enhancing the probability of yellow rust spores interacting with an internal sensor by a factor of between 20 and 25; demonstrating the effectiveness of the developed design. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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16 pages, 5663 KiB  
Article
Jet Oscillation Frequency Characterization of a Sweeping Jet Actuator
by Furkan Oz and Kursat Kara
Fluids 2020, 5(2), 72; https://doi.org/10.3390/fluids5020072 - 14 May 2020
Cited by 16 | Viewed by 3831
Abstract
The time-resolved flow field of a spatially oscillating jet emitted by a sweeping jet (SWJ) actuator is investigated numerically using three-dimensional Reynolds-averaged Navier–Stokes (3D-URANS) equations. Numerical simulations are performed for a range of mass flow rates providing flow conditions varying from incompressible to [...] Read more.
The time-resolved flow field of a spatially oscillating jet emitted by a sweeping jet (SWJ) actuator is investigated numerically using three-dimensional Reynolds-averaged Navier–Stokes (3D-URANS) equations. Numerical simulations are performed for a range of mass flow rates providing flow conditions varying from incompressible to subsonic compressible flows. After a detailed mesh study, the computational domain is represented using two million hexagonal control volumes. The jet oscillation frequency is predicted by analyzing velocity time histories at the actuator exit, and a linear relationship between the jet oscillation frequency and time-averaged exit nozzle Mach number is found ( f = 511.22   M + 46.618 , R² = 0.97). The results of our numerical model are compared with data from the literature, and a good agreement is found. In addition, we confirmed that the Strouhal number is almost constant with the Mach number for the subsonic oscillating jet and has an average value of St = 0.0131. The 3D-URANS model that we presented here provides a computationally inexpensive yet accurate alternative to the researchers to investigate jet oscillation characteristics. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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25 pages, 6446 KiB  
Article
On an Exact Step Length in Gradient-Based Aerodynamic Shape Optimization
by Farzad Mohebbi and Ben Evans
Fluids 2020, 5(2), 70; https://doi.org/10.3390/fluids5020070 - 13 May 2020
Cited by 1 | Viewed by 3270
Abstract
This study proposeda novel exact expression for step length (size) in gradient-based aerodynamic shape optimization for an airfoil in steady inviscid transonic flows. The airfoil surfaces were parameterized using Bezier curves. The Bezier curve control points were considered as design variables and the [...] Read more.
This study proposeda novel exact expression for step length (size) in gradient-based aerodynamic shape optimization for an airfoil in steady inviscid transonic flows. The airfoil surfaces were parameterized using Bezier curves. The Bezier curve control points were considered as design variables and the finite-difference method was used to compute the gradient of the objective function (drag-to-lift ratio) with respect to the design variables. An exact explicit expression was derived for the step length in gradient-based shape optimization problems. It was shown that the derived step length was independent of the method used for calculating the gradient (adjoint method, finite-difference method, etc.). The obtained results reveal the accuracy of the derived step length. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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19 pages, 2934 KiB  
Article
Multiscale Filtering of Compressible Wave Propagation in Complex Geometry through a Wavelet-Based Approach in the Framework of Pressurized Water Reactors Depressurization Transient Analysis
by Samy Mokhtari, Guillaume Ricciardi, Vincent Faucher, Pierre Argoul and Lucas Adélaide
Fluids 2020, 5(2), 64; https://doi.org/10.3390/fluids5020064 - 28 Apr 2020
Viewed by 2394
Abstract
The proposed research takes place in the framework of the analysis of the mechanical consequences of accidental scenarios for pressurized water reactors (PWR). It is particularly dedicated to the effects of the propagation of a transverse rarefaction wave through the assemblies of the [...] Read more.
The proposed research takes place in the framework of the analysis of the mechanical consequences of accidental scenarios for pressurized water reactors (PWR). It is particularly dedicated to the effects of the propagation of a transverse rarefaction wave through the assemblies of the nuclear core, consecutive to a pipe break in the primary circuit of the reactor. This paper focuses on the representation, with a reduced number of well-chosen variables, of a pressure wave propagating through a highly congested medium composed of rod bundles, with the primary objective of accurately evaluating the resulting pressure forces exerted on the rods. To achieve this goal, a description of the fluid domain as a homogenized or porous medium is introduced, yielding the need for a new filtering technique to be applied to the fluid fields. A new homogenized and multiscale representation of the fluid variables, based on continuous wavelet transform (CWT), is thus proposed. The capabilities of CWT to accurately approximate a reference representative unsteady pressure field, corresponding to a wave propagation at microscale, is assessed. The proposed technique is applied to a pressure field obtained numerically at local scale. The number of variables that shall be kept at macroscale to have a meaningful representation of the pressure field is fully evaluated through the comparison of the fluid force applied to the microstructure. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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16 pages, 656 KiB  
Article
Theoretical Background of the Hybrid VπLES Method for Flows with Variable Transport Properties
by Nikolai Kornev, Jordan Denev and Sina Samarbakhsh
Fluids 2020, 5(2), 45; https://doi.org/10.3390/fluids5020045 - 10 Apr 2020
Cited by 1 | Viewed by 2061
Abstract
The paper presents the theoretical basis for the extension of the V π LES method, originally developed in recent works of the authors for incompressible flows, to flows with variable density and transport properties but without chemical reactions. The method is based on [...] Read more.
The paper presents the theoretical basis for the extension of the V π LES method, originally developed in recent works of the authors for incompressible flows, to flows with variable density and transport properties but without chemical reactions. The method is based on the combination of grid based and grid free computational particle techniques. Large scale motions are modelled on the grid whereas the fine scale ones are modelled by particles. The particles represent the fine scale vorticity, and scalar quantities like e.g., temperature, mass fractions of species, density and mixture fraction. Coupled system of equations is derived for large and fine scales transport. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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18 pages, 8426 KiB  
Article
Numerical Simulation of Velocity Field around Two Columns of Tandem Piers of the Longitudinal Bridge
by Hongliang Qi, Junxing Zheng and Chenguang Zhang
Fluids 2020, 5(1), 32; https://doi.org/10.3390/fluids5010032 - 12 Mar 2020
Cited by 4 | Viewed by 2613
Abstract
This research explores the effects of different spans of two columns of tandem piers on the characteristics of x-velocity near the river bed based on computational fluid dynamics (CFD) simulations. With a span shorter than 27.5D (D is the diameter of piers), the [...] Read more.
This research explores the effects of different spans of two columns of tandem piers on the characteristics of x-velocity near the river bed based on computational fluid dynamics (CFD) simulations. With a span shorter than 27.5D (D is the diameter of piers), the shape and the lateral range of the x-velocity increases with the increase of distance downwards the x-direction. For the area between the tandem piers and the wall, the VRi/VR1 (the ratio of the x-velocity at the i-th row to the x-velocity of the first row in each model) near the wall increases up to 1.26. For the area between the two columns of tandem piers, the profile of VRi/VR1 changes from a “∩-shape” to an “M-shape” in each model. RAVC (average velocity change ratio) of different spans increases gradually and tends to be stable with the increases of the span. The largest RAVC is about −17.66% with a span of 0.52 m. The RMV (the ratio of the maximum x-velocity among piers in each row in different models to the maximum x-velocity of the two piers arranged side by side) of piers in the first row of different models is around 0.95. The RMV becomes 0.82 at the second pier in each model when the span is shorter than 27.5D, and increases to 0.91 if the span is longer than 27.5D. If the span is longer than 27.5D, the RMV of different piers are close to each other from the 2nd pier to the last one. Full article
(This article belongs to the Special Issue Recent Numerical Advances in Fluid Mechanics, Volume II)
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