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Publication date: 15 April 2024
Source: Computers & Fluids, Volume 273
Author(s): Nan Zhang
Publication date: 15 April 2024
Source: Computers & Fluids, Volume 273
Author(s):
Publication date: Available online 18 March 2024
Source: Computers & Fluids
Author(s): Niclas Jansson, Martin Karp, Artur Podobas, Stefano Markidis, Philipp Schlatter
Publication date: 15 April 2024
Source: Computers & Fluids, Volume 273
Author(s): Pratik Suchde, Heinrich Kraus, Benjamin Bock-Marbach, Jörg Kuhnert
Publication date: 15 April 2024
Source: Computers & Fluids, Volume 273
Author(s): Swapnil Soni, Avishek Ranjan, Trushar B. Gohil
Publication date: 15 April 2024
Source: Computers & Fluids, Volume 273
Author(s): Andrei Yu. Goldin, Shamil M. Magomedov, Luiz M. Faria, Aslan R. Kasimov
Publication date: 15 April 2024
Source: Computers & Fluids, Volume 273
Author(s): Kun Ye, Shubao Chen, Zhengyin Ye
Publication date: Available online 17 March 2024
Source: Computers & Fluids
Author(s): Pengxuan Luo, Jingxin Zhang
Publication date: Available online 16 March 2024
Source: Computers & Fluids
Author(s): Jiayi Cai, Pierre-Emmanuel Angeli, Jean-Marc Martinez, Guillaume Damblin, Didier Lucor
Publication date: Available online 4 March 2024
Source: Computers & Fluids
Author(s): T. Antritter, T. Josyula, T. Marić, D. Bothe, P. Hachmann, B. Buck, T. Gambaryan-Roisman, P. Stephan
In this article, the implicit-explicit RK time discretization coupled with LDG spatial discretization numerical algorithm for the micropolar fluid equations are presented. Moreover, the stability of the first-order and second-order fully discrete method are proved. Finally, the numerical experiments are given to verify the theoretical order and effectiveness of the presented method.
In this article, the spatial local discontinuous Galerkin (LDG) approximation coupled with the temporal implicit-explicit Runge–Kutta (RK) evolution for the micropolar fluid equations are adopted to construct the discretization method. To avoid the incompressibility constraint, the artificial compressibility strategy method is used to convert the micropolar fluid equations into the Cauchy–Kovalevskaja type equations. Then the LDG method based on the modal expansion and the implicit-explicit RK method are properly combined to construct the expected third-order method. Theoretically, the unconditionally stable of the fully discrete method are derived in multidimensions for triangular meshs. And the numerical experiments are given to verify the theoretical and effectiveness of the presented methods.
We propose in this article a discretization of the momentum convection operator for the approximation of the Navier–Stokes equations by the low-order nonconforming Rannacher–Turek finite element. This operator is of finite volume type, and its almost second order expression is derived by an algebraic MUSCL-like technique; it satisfies a discrete and local kinetic energy conservation identity. The stability, consistency, and accuracy of the resulting scheme is assessed by numerical experiments for incompressible, barotropic and compressible flows, including the Euler equations.
We propose in this article a discretization of the momentum convection operator for fluid flow simulations on quadrangular or generalized hexahedral meshes. The space discretization is performed by the low-order nonconforming Rannacher–Turek finite element: the scalar unknowns are associated with the cells of the mesh while the velocities unknowns are associated with the edges or faces. The momentum convection operator is of finite volume type, and its expression is derived, as in MUSCL schemes, by a two-step technique: (i)$$ (i) $$ computation of a tentative flux, here, with a centered approximation of the velocity, and (ii)$$ (ii) $$ limitation of this flux using monotonicity arguments. The limitation procedure is of algebraic type, in the sense that its does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows and may thus be implemented in a scheme for incompressible or compressible flows. To achieve this goal, we derive a discrete analogue of the computation ui(∂t(ρui)+div(ρuiu)=12∂t(ρui2)+12div(ρui2u)$$ {u}_i\kern0.3em \Big({\partial}_t\left(\rho {u}_i\right)+\operatorname{div}\left(\rho {u}_i\boldsymbol{u}\right)=\frac{1}{2}{\partial}_t\left(\rho {u}_i^2\right)+\frac{1}{2}\operatorname{div}\left(\rho {u}_i^2\boldsymbol{u}\right) $$ (with u$$ \boldsymbol{u} $$ the velocity, ui$$ {u}_i $$ one of its component, ρ$$ \rho $$ the density, and assuming that the mass balance holds) and discuss two applications of this result: first, we obtain stability results for a semi-implicit in time scheme for incompressible and barotropic compressible flows; second, we build a consistent, semi-implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy. The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier–Stokes equations, the barotropic and the full compressible Navier–Stokes equations and the compressible Euler equations.
We are interested in the efficient solution of the saddle-point linear systems arising from isogeometric analysis approach of Navier–Stokes equations. We use a preconditioned Krylov subspace method (GMRES) with efficient approximate solvers for solving subsystems within preconditioning. We investigate the impact on the convergence of the outer solver and aim to identify an effective combination preconditioners and inner solvers with respect problem parameters.
We deal with efficient numerical solution of the steady incompressible Navier–Stokes equations (NSE) using our in-house solver based on the isogeometric analysis (IgA) approach. We are interested in the solution of the arising saddle-point linear systems using preconditioned Krylov subspace methods. Based on our comparison of ideal versions of several state-of-the-art block preconditioners for linear systems arising from the IgA discretization of the incompressible NSE, suitable candidates have been selected. In the present paper, we focus on selecting efficient approximate solvers for solving subsystems within these preconditioning methods. We investigate the impact on the convergence of the outer solver and aim to identify an effective combination. For this purpose, we compare convergence properties of the selected solution approaches for problems with different viscosity values, mesh refinement levels and discretization bases.
For challenging volume-of-fluid based simulations of turbulent bubbly flows characterized by high Reynolds numbers, significant bubble deformations and large-eddy-simulation-like grid resolutions, severe volume errors for the gas phase can accumulate. To avoid this, two methods to accurately conserve the gas volume are proposed, which allow to perform numerical simulations that have previously been out of reach for this numerical framework. Their reliability is demonstrated based on an investigation of global flow characteristics and individual bubble dynamics.
This study proposes two different strategies to enforce accurate volume conservation in volume-of-fluid (VOF)-based simulations of turbulent bubble-laden flows on coarse grids. It is demonstrated that, without a correction, minimal volume errors on a time-step level, caused by the under-resolution of the interface, can accumulate to significant deviations from the intended flow conditions despite the comparably good volume conservation properties of the geometric VOF method. In particular, large volume errors are observed for challenging setups combining coarse grid resolutions and comparably high Reynolds and Eötvös numbers. The problem is reinforced for long-term simulations in periodic domains, which are often performed to collect flow statistics of bubbly flows. The first proposed volume conservation method simply corrects the volume error of a bubble by uniformly adding or removing the respective amount of gas volume in the interface cells. The second proposed method performs an additional reconstruction and advection step of the VOF field using a non-divergence-free velocity field, which can be interpreted as a slight dilatation or contraction of the bubble. A comparison between the global flow statistics as well as the individual bubble dynamics for both volume conservation methods reveals that the results are quasi-identical for a number of challenging test cases, while the gas volume is accurately conserved. The proposed methods allow to perform numerical simulations of freely deformable bubbles in turbulent flows for setups that have previously been out of reach for this numerical framework.
We propose two enhanced approaches of physics informed neural networks (PINN) for solving the challenging Navier–Stokes equation (NSE). The first approach improves the model by approximating the velocity components and integrating a pressure-based loss function. The second approach directly approximates the NSE solution without assumptions, significantly reducing training duration while maintaining high accuracy. We successfully apply this approach to solve the three-dimensional NSE, demonstrating the advantages of our models in terms of trainability, flexibility, and efficiency.
Fluid mechanics is a critical field in both engineering and science. Understanding the behavior of fluids requires solving the Navier–Stokes equation (NSE). However, the NSE is a complex partial differential equation that can be challenging to solve, and classical numerical methods can be computationally expensive. In this paper, we propose enhancing physics-informed neural networks (PINNs) by modifying the residual loss functions and incorporating new computational deep learning techniques. We present two enhanced models for solving the NSE. The first model involves developing the classical PINN for solving the NSE, based on a stream function approach to the velocity components. We have added the pressure training loss function to this model and integrated the new computational training techniques. Furthermore, we propose a second, more flexible model that directly approximates the solution of the NSE without making any assumptions. This model significantly reduces the training duration while maintaining high accuracy. Moreover, we have successfully applied this model to solve the three-dimensional NSE. The results demonstrate the effectiveness of our approaches, offering several advantages, including high trainability, flexibility, and efficiency.
A novel coupled Euler–Lagrange method, arranging Lagrange particles in the Euler grid to track shock front and discontinuities, is proposed to capture shock waves with large deformation. It solves over-stretching problem of particles with large deformation and improve the mapping accuracy by the weighted mutual mapping method. Virtual particle method realizes flow boundary conditions easily and maintains accuracy level of boundary.
The accurate capturing of shock waves by numerical methods has long been a focus of attention in engineering owing to singularity problems in discontinuities. In this article, a novel coupled Euler–Lagrange method (CELM) is proposed to capture shock waves and discontinuities with high resolution and high order of mapping accuracy. CELM arranges the Lagrange particles on an Euler grid to track the discontinuous points automatically, and the data pertaining to the grids and particles interact via a weighted mutual mapping method that not only achieves fourth-order accuracy in a smooth area of the solution but also maintains a steep discontinuous transition in the discontinuous area. In the virtual particle method, virtual particles are derived from the existing real particles; thus, the inflow and outflow of the particles and interpolation accuracy of the boundary are more easily realized. An accuracy test and energy convergence test demonstrated the fourth-order convergence accuracy and low energy dissipation of the CELM; the method exhibited lower error and better conservation ability than high-precision schemes such as WENO3 and WENO5. The Sod shock tube problem and Woodward–Colella problem showed higher discontinuity resolution of the CELM and ability to accurately track discontinuity points. Examples of Riemann problems were employed to prove that the CELM exhibits lower dissipation and higher shock resolution than WENO3 and WENO5. The CELM also showed an accurate structure based on particle distribution. Shockwave diffraction tests were conducted to prove that the CELM results showed good agreement with the experimental data and exhibited an accurate expansion wave. The CELM can also accurately simulate the collision of an expansion wave and vortex.
We propose a general vorticity-streamfunction formulation, together with redefined vorticity and streamfunction, for binary fluids with an arbitrary density contrast, where the interface is captured using the phase-field model based on the conservative Allen-Cahn equation. Numerical tests are conducted under the framework of Lattice Boltzmann method, including the Capillary wave, the Rayleigh–Taylor instability, and the droplet splashing on a thin film, showing good agreements with analytical predictions and enforced incompressibility of fluids.
The classical vorticity-streamfunction formulation (VSF) can avoid the difficulty in the calculation of pressure gradient term of the Navier Stokes equation via eliminating pressure gradient term from the theoretical basis. Within this context we propose a general VSF, together with redefined vorticity and streamfunction, so as to realize numerically stable and reliable simulations of binary fluids with an arbitrary density contrast. By incorporating the interface-tracking phase-field model based on the conservative Allen-Cahn equation [Phys. Rev. E 94, 023311 (2016)], the binary flow simulation framework is established. Numerical tests are conducted using the Lattice Boltzmann method (LBM), which is usually regarded as an easy-to-use tool for solving the Navier–Stokes equation but generally suffers from the drawback of not being capable of enforcing incompressibility. The LBM herein functions as a numerical tool for solving the vorticity transport equation, the streamfunction equation, and the conservative Allen-Cahn equation. Three two-dimensional benchmark cases, i.e., the Capillary wave, the Rayleigh–Taylor instability, and the droplet splashing on a thin liquid film, are discussed in detail to verify the present methodology. Results show good agreements with both analytical predictions and literature data, as well as good numerical stability in terms of high density ratio and high Reynolds number. Overall, the general VSF inherits the intrinsic superiority of the classical VSF in enforcing incompressibility, and offers a useful and reliable alternative for binary flow modeling.
By combining with a novel decoupling technique based on the “zero-energy-contribution” feature and the pressure correction method, the linearized second order BDF numerical scheme, which has the advantages of fully decoupled structure, is constructed. The unconditional energy stability of the scheme is strictly proved and the detailed implementation process that only requires to calculate several linear elliptic equations with constant coefficients is given.
In this paper, we analyze the numerical approximation of two-phase magnetohydrodynamic flows. Firstly, an equivalent new system is designed by introducing two scalar auxiliary variables. One of variables is used to linearize the phase field function and the other is used to deal with the highly coupled and nonlinear terms. Secondly, by combining with a novel decoupling technique based on the “zero-energy-contribution” feature and the pressure correction method, the linearized second order BDF numerical scheme, which has the advantage of fully decoupled structure, is constructed. Furthermore, we strictly prove the unconditional energy stability and error analysis of the scheme, and give a detailed implementation procedure that only requires to calculate several linear elliptic equations with constant coefficients. Finally, the results of numerical simulations are presented to validate the rates of convergence and energy stability.
Convergence histories of the GMRES+AMG solver with a polynomial Guass-Seidel and the ILU direct and iterative smoothers, for a matrix from the projection step in PeleLM with 11M unknowns, solved on NREL's Eagle computer. The iterative ILU smoother shows a 5X speed-up over the direct smoother.
Incomplete LU (ILU) smoothers are effective in the algebraic multigrid (AMG) V$$ V $$-cycle for reducing high-frequency components of the error. However, the requisite direct triangular solves are comparatively slow on GPUs. Previous work has demonstrated the advantages of Jacobi iteration as an alternative to direct solution of these systems. Depending on the threshold and fill-level parameters chosen, the factors can be highly nonnormal and Jacobi is unlikely to converge in a low number of iterations. We demonstrate that row scaling can reduce the departure from normality, allowing us to replace the inherently sequential solve with a rapidly converging Richardson iteration. There are several advantages beyond the lower compute time. Scaling is performed locally for a diagonal block of the global matrix because it is applied directly to the factor. Further, an ILUT Schur complement smoother maintains a constant GMRES iteration count as the number of MPI ranks increases, and thus parallel strong-scaling is improved. Our algorithms have been incorporated into hypre, and we demonstrate improved time to solution for linear systems arising in the Nalu-Wind and PeleLM pressure solvers. For large problem sizes, GMRES+$$ + $$AMG executes at least five times faster when using iterative triangular solves compared with direct solves on massively parallel GPUs.
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Nadir-Alexandre Messaï, Guillaume Daviller, Jean-François Boussuge
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Yi Zhao, Dongting Cai, Junxiang Yang
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Thomas Bellotti
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Hendrik Fischer, Julian Roth, Thomas Wick, Ludovic Chamoin, Amelie Fau
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Zhihang Xu, Qifeng Liao, Jinglai Li
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Robert Marskar
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Thomas Bonnafont, Delphine Bessieres, Jean Paillol
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Xiaokai Yuan, Peijun Li
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Rui Gao, Indu Kant Deo, Rajeev K. Jaiman
Publication date: 1 May 2024
Source: Journal of Computational Physics, Volume 504
Author(s): Chunheng Zhao, Jacob Maarek, Seyed Mohammadamin Taleghani, Stephane Zaleski
The stability of the flow past a circular cylinder in the presence of a wavy ground is investigated numerically in this paper. The wavy ground consists of two complete waves with a wavelength of 4D and an amplitude of 0.5D, where D is the cylinder diameter. The vertical distance between the cylinder and the ground is varied, and four different cases are considered. The stability analysis shows that the critical Reynolds number increases for cases close to the ground when compared to the flow past a cylinder away from the ground. The maximum critical Reynolds number is obtained when the cylinder is located in front of the waves. The wavy ground adds layers of clockwise (negative) vorticity due to flow separation from the wave peak, to the oscillating Kármán vortex. This negative vorticity from the wave peak also cancels part of the positive (counterclockwise) vorticity shed from the bottom half of the cylinder. In addition, the negative vorticity from the wave peak strengthens the clockwise (negative) vorticity shed from the top half of the cylinder. These interactions combined with the ground effect skewed the flow away from the ground. The base flow is skewed upward for all the near-ground cases. However, this skew is larger when the cylinder is located over the wavy ground. The vortex shedding frequency is also altered due to the presence of the waves. The main eigenmode found for plain flow past a cylinder appears to become suppressed for cases closer to the ground. Limited particle image velocimetry experiments are reported which corroborate the finding from the stability analysis.
An inviscid vortex shedding model is numerically extended to simulate falling flat plates. The body and vortices separated from the edge of the body are described by vortex sheets. The vortex shedding model has computational limitations when the angle of incidence is small and the free vortex sheet approaches the body closely. These problems are overcome by using numerical procedures such as a method for a near-singular integral and the suppression of vortex shedding at the plate edge. The model is applied to a falling plate of flow regimes of various Froude numbers. For \(\text {Fr}=0.5\) , the plate develops large-scale side-to-side oscillations. In the case of \(\text {Fr}=1\) , the plate motion is a combination of side-to-side oscillations and tumbling and is identified as a chaotic type. For \(\text {Fr}=1.5\) , the plate develops to autorotating motion. Comparisons with previous experimental results show good agreement for the falling pattern. The dependence of change in the vortex structure on the Froude number and its relation with the plate motion is also examined.
A theoretical and experimental study was conducted to investigate the effect of injection angle on surface waves. Linear stability theory was utilized to obtain the analytical relation. In the experimental study, high-speed photography and shadowgraph techniques were used. Image processing codes were developed to extract information from photos. The results obtained from the theoretical relation were validated with the experimental results at different injection angles. In addition, at the injection angle of 90 \({^\circ }\) , the theoretical results were evaluated with the experimental results of other researchers. This evaluation showed that the theory results were in good agreement with the experimental data. The proper orthogonal decomposition (POD) and the power spectra density (PSD) analysis were also used to investigate the effect of the injection angle on the flow structures. The results obtained from the linear stability were used to determine the maximum waves’ growth rate, and a relation was presented for the breakup length of the liquid jet at different injection angles. The breakup length results were compared with theory and published experimental data. The presented relation is more consistent with experimental data than other theories due to considering the nature of waves. The results showed that the instability of the liquid jet is influenced by three forces: inertial, surface tension, and aerodynamic. Therefore, Rayleigh–Taylor, Kelvin–Helmholtz, Rayleigh–Plateau, and azimuthal instabilities occur in the process. Decreasing the injection angle changes the nature of waves and shifts from Rayleigh–Taylor to Kelvin–Helmholtz. That reduces the wavelength and increases the growth rate of the waves. Axial waves have a significant impact on the physics of the waves and influence parameters. If axial waves are not formed, the growth rate of the waves is independent of the injection angle. An increase in the gas Weber number causes a change in the type of dominant waves and a greater instability of the liquid jet. In contrast, an increase in the liquid Weber number causes an enhancement in the resistance of the liquid jet against the transverse flow without changing the type of the dominant waves. Decreasing the density ratio reduces the effect of Rayleigh–Taylor waves and strengthens the Kelvin–Helmholtz waves. It causes two trends to be observed for the growth rate of waves at low spray angles, while one trend occurs at high spray angles.
The focus here is on a thin solid body passing through a channel flow and interacting with the flow. Unsteady two-dimensional interactive properties from modelling, analysis and computation are presented along with comparisons. These include the effects of a finite dilation or constriction, as the body travels through, and the effects of a continuing expansion of the vessel. Finite-time clashing of the body with the channel walls is investigated as well as the means to avoid clashing. Sustained oscillations are found to be possible. Wake properties behind the body are obtained, and broad agreement in trends between full-system and reduced-system responses is found for increased body mass.
Recently, Rim (Ocean Engng 239:711, 2021; J Ocean Engng Mar Energy 9:41-51, 2023 ) suggested an exact DtN artificial boundary condition to study the three-dimensional wave diffraction by stationary bodies. This paper is concerned with three-dimensional linear interaction between a submerged oscillating body with arbitrary shape and the regular water wave with finite depth. An exact Dirichlet-to-Neumann (DtN) boundary condition on a virtual cylindrical surface is derived, where the virtual surface is chosen so as to enclose the body and extract an interior subdomain with finite volume from the horizontally unbounded water domain. The DtN boundary condition is then applied to solve the interaction between the body and the linear wave in the interior subdomain by using boundary integral equation. Based on verification of the present model for a submerged vertical cylinder, the model is extended to the case of a submerged chamfer box with fillet radius in order to study 6-DoF oscillatory motion of the body under the free surface wave.
The effect of non-Boltzmann energy distributions on the free propagation of shock waves through a monoatomic gas is investigated via theory and simulation. First, the non-Boltzmann heat capacity ratio \(\gamma \) , as a key property for describing shock waves, is derived from first principles via microcanonical integration. Second, atomistic molecular dynamics simulations resembling a shock tube setup are used to test the theory. The presented theory provides heat capacity ratios ranging from the well-known \(\gamma = 5/3\) for Boltzmann energy-distributed gas to \(\gamma \rightarrow 1\) for delta energy-distributed gas. The molecular dynamics simulations of Boltzmann and non-Boltzmann driven gases suggest that the shock wave propagates about 9% slower through the non-Boltzmann driven gas, while the contact wave appears to be about 4% faster if it trails non-Boltzmann driven gas. The observed slowdown of the shock wave through applying a non-Boltzmann energy distribution was found to be consistent with the classical shock wave equations when applying the non-Boltzmann heat capacity ratio. These fundamental findings provide insights into the behavior of non-Boltzmann gases and might help to improve the understanding of gas dynamical phenomena.
Under the low-noise Mach 3 flight conditions for a supersonic passenger aircraft having unswept wings with a thin parabolic airfoil, laminar-turbulent transition is due to amplification of the first mode. Stability of a local self-similar boundary layer over such a wing is investigated both using the \(e^{N}\) method in the framework of linear stability theory and direct numerical simulation (DNS). It is found that the instability amplitude should reach a maximum over the entire spectral range above the profiles of 2.5% and thicker. The locus of maximum appears at the trailing edge and moves to the leading edge as the profile becomes thicker, while the maximum amplitude decreases. The theoretical findings are supported by DNS of the linear wave packets propagating in the boundary layer. Significance of these results to the design of laminar supersonic wings is discussed.
An adjoint-based method is presented for determining manufacturing tolerances for aerodynamic surfaces with natural laminar flow subjected to wavy excrescences. The growth of convective unstable disturbances is computed by solving Euler, boundary layer, and parabolized stability equations. The gradient of the kinetic energy of disturbances in the boundary layer (E) with respect to surface grid points is calculated by solving adjoints of the governing equations. The accuracy of approximations of \(\Delta E\) , using gradients obtained from adjoint, is investigated for several waviness heights. It is also shown how second-order derivatives increase the accuracy of approximations of \(\Delta E\) when surface deformations are large. Then, for specific flight conditions, using the steepest ascent and the sequential least squares programming methodologies, the waviness profile with minimum \(L2-\) norm that causes a specific increase in the maximum value of N- factor, \(\Delta N\) , is found. Finally, numerical tests are performed using the NLF(2)-0415 airfoil to specify tolerance levels for \(\Delta {N}\) up to 2.0 for different flight conditions. Most simulations are carried out for a Mach number and angle of attack equal to 0.5 and \(1.25^{\circ }\) , respectively, and with Reynolds numbers between \(9\times 10^6\) and \(15\times 10^6\) and for waviness profiles with different ranges of wavelengths. Finally, some additional studies are presented for different angles of attack and Mach numbers to show their effects on the computed tolerances.
A low-order physics-based model to simulate the unsteady flow response to airfoils undergoing large-amplitude variations of the camber is presented in this paper. Potential-flow theory adapted for unsteady airfoils and numerical methods using discrete-vortex elements are combined to obtain rapid predictions of flow behavior and force evolution. To elude the inherent restriction of thin-airfoil theory to small flow disturbances, a time-varying chord line is proposed in this work over which to satisfy the appropriate boundary condition, enabling large deformations of the camber line to be modeled. Computational fluid dynamics simulations are performed to assess the accuracy of the low-order model for a wide range of dynamic trailing-edge flap deflections. By allowing the chord line to rotate with trailing-edge deflections, aerodynamic loads predictions are greatly enhanced as compared to the classical approach where the chord line is fixed. This is especially evident for large-amplitude deformations.
Flickering buoyant diffusion methane flames in weakly rotatory flows were computationally and theoretically investigated. The prominent computational finding is that the flicker frequency nonlinearly increases with the nondimensional rotational intensity R (up to 0.24), which is proportional to the nondimensional circumferential circulation. This finding is consistent with the previous experimental observations that rotatory flows enhance flame flicker to a certain extent. Based on the vortex-dynamical understanding of flickering flames that the flame flicker is caused by the periodic shedding of buoyancy-induced toroidal vortices, a scaling theory is formulated for flickering buoyant diffusion flames in weakly rotatory flows. The theory predicts that the increase of flicker frequency f obeys the scaling relation \(\left( f-f_{0} \right) \propto R^{2}\) , which agrees very well with the present computational results. In physics, the external rotatory flow enhances the radial pressure gradient around the flame, and the significant baroclinic effect \(\mathrm {\nabla }p\times \mathrm {\nabla }\rho \) contributes an additional source for the growth of toroidal vortices so that their periodic shedding is faster.