Navier-Stokes Flow Research Articles

TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method

In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains.

Time Evolution of the Navier–Stokes Flow in Far-Field

Time Evolution of the Navier–Stokes Flow in Far-Field

Particle response to oscillatory flows at finite Reynolds numbers

The response of spherical particles to oscillatory fluid flow forcing at finite Reynolds numbers exhibits significant deviations from classical analytical predictions due to nonlinear convective contributions. This study employs finite element simulations to explore the long-term (stationary) behavior of such particles across a wide range of conditions, including various external and particle Reynolds numbers, Strouhal numbers, and fluid-to-particle density ratios. Key contributions of this work include determining the range of validity of Tchen's equation of motion for infinitesimal and finite Reynolds numbers and correlating particle response for a wide range of density ratios and flow conditions at high frequency oscillations. This work introduces a modified form of the history drag term in a newly proposed Lagrangian equation of motion. The new equation incorporates a parameter-dependent fractional-order derivative tailored to accommodate nonlinearities due to convective effects. These novel correlations not only extend the operational range of existing model equations but also provide accurate estimates of particle response under a range of external flow conditions, as validated by comparison with numerical solutions of the Navier–Stokes flow around the particles.

Analysis of the aerothermal performance of modern commercial high-pressure turbine rotors using different levels of fidelity

In the field of design and optimization of sophisticated geometries such as film-cooled turbine blades of modern jet engines, Computational Fluid Dynamics (CFD) simulation is commonly employed. As the pursuit for higher turbine entry temperatures intensifies, it becomes crucial for computational analyses to offer accurate predictions of metal temperatures and heat transfer coefficients on these critical components. This study investigates the impact of key physical factors that characterize the operation of these components on the accuracy of the computational model. In particular, the effects of including the exchange of thermal energy between the fluid and solid domains, as well as modelling the unsteady interaction between the rotor and stator are studied. The research focuses on a fully featured one-and-a-half stage high-pressure turbine of a commercial jet engine, utilizing proprietary software for conducting 3D Reynolds-Averaged Navier-Stokes flow simulations. To model the fluid-solid thermal interaction, steady-state Conjugate Heat Transfer (CHT) simulations are performed. The CHT results are then compared with experimental data obtained from a thermal paint test, achieving good levels of agreement. Additionally, phase-lag simulations are executed under adiabatic-wall conditions to evaluate the influence of stator-rotor interaction on near-wall gas temperatures. This work shows that, by modelling the periodic unsteadiness at the stator-rotor interface with a phase-lag technique, the maximum near-wall gas temperature prediction is significantly increased, sometimes more than 100K, with respect to the steady-state model one. Findings from this work also suggest that simplified “strip” source terms used to model the presence of film cooling hole rows on the surface of a blade can be used with satisfactory accuracy only for nominal conditions. The mass flows delivered by each source strip should not be linearly scaled based on mainstream quantities for use in different conditions, but they should be recalculated from scratch for the new conditions.

Almost periodic motions and their stability of the non-autonomous Oseen–Navier–Stokes flows

Almost periodic motions and their stability of the non-autonomous Oseen–Navier–Stokes flows

Remarks on the separation of Navier–Stokes flows

Recently, strong evidence has accumulated that some solutions to the Navier–Stokes equations in physically meaningful classes are not unique. The primary purpose of this paper is to establish necessary properties for the error of hypothetical non-unique Navier–Stokes flows under conditions motivated by the scaling of the equations. Our first set of results show that some scales are necessarily active—comparable in norm to the full error—as solutions separate. ‘Scale’ is interpreted in several ways, namely via algebraic bounds, the Fourier transform and discrete volume elements. These results include a new type of uniqueness criteria which is stated in terms of the error. The second result is a conditional predictability criteria for the separation of small perturbations. An implication is that the error necessarily activates at larger scales as flows de-correlate. The last result says that the error of the hypothetical non-unique Leray–Hopf solutions of Jia and Šverák locally grows in a self-similar fashion. Consequently, within the Leray–Hopf class, energy can hypothetically de-correlate at a rate which is faster than linear. This contrasts numerical work on predictability which identifies a linear rate. Our results suggest that this discrepancy may be explained by the fact that non-uniqueness might arise from perturbation around a singular flow.

A multiblock (MIB) finite element method for accurate and efficient blood flow simulation

We present an efficient and accurate immersed boundary (IB) finite element (FE) method for internal flow problems with complex geometries (e.g. blood flow in the vascular system). In this study, we use a voxelized flow domain (discretized with hexahedral and tetrahedral elements) instead of a box domain, which is frequently used in IB methods. We highlight the applicability and efficiency of the proposed method in generating the flow domain mesh and removing unnecessary portions of the domain, compared to the standard IB methods that rely on a full Cartesian mesh. Our approach is attractive for internal flow problems that use IB methods. The proposed method accurately satisfies the boundary conditions on the immersed object using velocity and pressure correction procedures. The velocity correction applies implicitly such that the velocity on the immersed boundary (Lagrangian points) interpolated from the corrected velocity values computed on the mesh nodes (Eulerian nodes) accurately satisfies the prescribed velocity boundary conditions. The proposed method utilizes the well-established incremental pressure correction scheme (IPCS) FE solver, and the boundary condition-enforced IB (BCE-IB) method to numerically solve the transient, incompressible Navier-Stokes (NS) flow equations. We verify the accuracy of our numerical method using the analytical solution for the Poiseuille flow in a cylinder, and the available experimental data (laser Doppler velocimetry) for the flow in a three-dimensional 90° angle tube bend. We further examine the accuracy and applicability of the proposed method by considering flow within complex geometries, such as blood flow in aneurysmal vessels and the aorta, flow configurations that would otherwise be difficult to solve by most IB methods. Our method offers high accuracy, as demonstrated by the verification examples, and high applicability, as demonstrated through the solution of blood flow within complex geometry. The proposed method is efficient, since it is as fast as the traditional finite element method used to solve the Navier–Stokes flow equations, with a small overhead (not more than 5%) due to the numerical solution of a linear system formulated for the IB method.

Unraveling hydrodynamic interactions in fish schools: A three-dimensional computational study of in-line and side-by-side configurations

We numerically investigate the hydrodynamic interactions between a pair of three-dimensional (3D) fish-like bodies arranged in both in-line and side-by-side configurations. The morphology and kinematics of these fish-like bodies are modeled on a live rainbow trout (Oncorhynchus mykiss) observed during steady swimming in the laboratory. An immersed-boundary-method-based incompressible Navier–Stokes flow solver is employed to capture the flow dynamics around the fish-like bodies accurately. Our findings indicate that hydrodynamic performance of individual fish in both arrangements is influenced by their spatial separation when in close proximity as well as by the relative phase difference between the two fish. In the case of in-phase in-line schools, the leading fish experiences up to 5.3% increase in propulsive efficiency, attributed to the water blockage effect caused by the following fish. In comparison, the following fish experiences an increase in drag and power consumption along its body. Detailed analysis reveals that this rise in drag primarily results from an increase in friction drag (89%), driven by the amplified velocity field around the fish's body. Furthermore, altering the phase difference between the fish can help reduce pressure drag on the following fish by affecting the interaction between incoming vortex rings and its trunk. In side-by-side schools with in-phase swimming, a reduction of 6.8% in power consumption on the caudal fin is achieved for each fish when the transverse distance is maintained at 0.25 body lengths. Flow analysis reveals that the decrease in power usage is attributed to a diminished velocity field between the caudal fins, facilitating flow separation and subsequently reducing energy expenditure required for generating comparative thrust. For the out-of-phase swimming, the side-by-side school system experiences enhanced thrust production, owing to a wake energy recapture mechanism. The degree of enhancement varies for each fish and is determined by the specific phase difference. These insights obtained from our study hold the potential to inform the design and navigation strategies of underwater robotic swarms.

Data-driven stabilized finite element solution of advection-dominated flow problems

In this article, we address the solution of advection-dominated flow problems by stabilized methods, by means of data-driven least-squares computed stabilized coefficients. As main methodological tool, we introduce a data-driven off-line/on-line strategy to compute them with low computational cost.We compare the errors provided by the data-driven stabilized coefficients to those provided by several previously established stabilized coefficients within the solution of advection–diffusion and Navier–Stokes flows, on structured and un-structured grids, with Lagrange Finite Elements up to third degree of interpolation. We obtain substantial error improvements for high-order finite element interpolation.We conclude that the data-driven procedure is a rewarding procedure, worth to be applied to general stabilized solutions of general flow problems.

Development and calibration of a fast flow model for solid oxide cell stack internal manifolds

Due to the commercialization of solid oxide cells (SOC) progressing at an accelerated pace, computationally inexpensive SOC models adapted to the iterative nature of the engineering process are in increasing demand. Flow simulation in the stack is especially challenging in this regard because detailed computational fluid mechanic models are computationally demanding, while simplified models rely on pressure loss coefficients and friction factors not readily available in the literature. In this study, a computationally inexpensive algebraic model of an SOC stack internal manifold is developed and calibrated for laminar flow conditions. Thereby, pressure loss coefficients and Darcy friction factors are determined for a broad range of operating conditions through symbolic regression of Navier–Stokes flow simulation results of stacks of 20 to 40 cells. The derived Darcy friction factors for the inlet and outlet manifolds prove to be of particular importance, as they deviate strongly from the expressions assumed in similar modeling studies. The predictive power of the developed model is demonstrated by providing accurate predictions of the flow distribution in the stack, even outside of the calibration window.

Equilibrium Configurations of a Symmetric Body Immersed in a Stationary Navier–Stokes Flow in a Planar Channel

Equilibrium Configurations of a Symmetric Body Immersed in a Stationary Navier–Stokes Flow in a Planar Channel

Numerical investigation of vortex-induced forces on a wind turbine tower segment at very high Reynolds numbers

The vortex-induced forces on an extruded cylinder with a span of two diameters representative of a finite segment of a non-tapered wind turbine tower at a very high Reynolds number (Re = 8.0×106) are numerically investigated using an incompressible Navier-Stokes flow solver with an Improved Delayed Detached Eddy Simulation (IDDES) turbulence model and correlation-based boundary layer transition modelling. The solution shows spanwise correlated structured vortex shedding with the Strouhal number St = 0.48. The boundary layer transition is found to occur at θ transition = 70 ◦ , and boundary layer separation is found to occur at θ separation = 120 ◦ . Results from the grid dependency study strongly imply that when using IDDES, the Strouhal number converges to higher values than previously reported by the literature as the grid is refined, with results ranging from St ∼ 0.44 using a grid with 4.2 × 106 cells, to St = 0.48 for the finest considered grid with 33 × 106 cells. This behaviour is not seen for URANS, where St = 0.33 for the finest grid.

Axial Green Function Method in an Efficient Projection Scheme for Incompressible Navier–Stokes Flow in Arbitrary Domains

We introduce a new approach to solving incompressible Navier–Stokes flow. This method combines a projection scheme with the Axial Green Function Method (AGM). Based on the Kim and Moin methods, our methodology employs a predictor–corrector mechanism to achieve stable and accurate velocity corrections. Using axial Green functions, we transform complex differential equations into simpler one-dimensional integral equations. These are strategically placed along a minimal axis-parallel lines, known as axial lines, within the flow domain. This transformation makes computation and analysis more efficient from a numerical viewpoint. A significant innovation in our approach is using one-dimensional axial Green functions tailored explicitly for the reaction–diffusion ordinary differential operator. These functions efficiently handle the discrete-time derivative and viscous terms of the momentum equation. Furthermore, our approach allows for the arbitrary construction of axis-parallel lines, facilitating analysis near critical flow regions and even enabling the random distribution of these lines. We validate our proposed method through numerical examples, demonstrating the convergence of numerical solutions, the effectiveness of arbitrarily constructed axis-parallel lines, and the potential extension of our method to three-dimensional flow problems. Additionally, this study provides a robust and adaptable alternative way to solve incompressible Navier–Stokes flows, proving its effectiveness in practical applications such as Tesla valves.

A pressure-projection pre-conditioning multi-fractional-step method for Navier–Stokes Flow in Porous Media

A new pressure-projection pre-conditioning multi-fractional-step (PPP-MFS) method for incompressible Navier–Stokes flow in porous media is presented. This fractional step method is applied to decouple the pressure and the velocity; thereby, overcoming the computational costs and difficulty associated with the resolution of the nonlinear term in the Navier–Stokes equation for fine geometric models. Specifically, time evolution is decomposed into a sequence of multi-fractional solution steps. In the first step, an elliptic problem is solved for pressure (p) with a no-slip boundary condition. This gives the Stokes pressure and velocity fields. In the second step, the obtained pressure p is then projected onto the field p∗ and used to solve for velocity field (u) required for the pre-conditioning of the solution to the Navier–Stokes equation. The pressure and velocity fields are obtained from the solution of the Navier–Stokes equation in the third step. Numerical and geometric discretisation of porous samples were carried out using finite-element method. For flow in simple channel models represented by two- and three-dimensions and in systems with high conductivity, the Stokes and Navier–Stokes numerical solutions produced close pressure and velocity field approximations. For flow around a cylinder, computation time is consistently higher in the Navier–Stokes equation by a factor of 2 with a pronounced non-symmetric pressure field at high mesh refinements. This computation time is desirable given that Navier–Stokes computation without preconditioning can be orders of magnitude more expensive.

On the stability of stationary compressible Navier–Stokes flows in 3D

AbstractThis paper studies the stability of the stationary solution of the compressible Navier–Stokes equation in the 3D whole space with an external force which decays at spatial infinity. We obtain the global existence result of the non-stationary problem under the smallness assumptions on the initial perturbation around the small stationary solution. We also derive the time decay rates of the perturbations under the smallness assumption on the initial perturbations, and show the optimality of the time decay rates. The proofs are based on the combination of the spectral analysis and energy method in the framework of Besov spaces. The time-space integral estimate for the linearized semigroup around the constant state in some Besov spaces plays a crucial role in the proofs.

Topology optimization of steady Navier-Stokes flow using moving morphable void method

In order to explicitly and smoothly implement fluid-based topology optimization for improving the performance and manufacturing efficiency of flow systems, the moving morphable void (MMV) method is employed to optimize the steady-state navier-stokes (NS) flow for minimizing the dissipation energy under a volume constraint. To this end, a MMV-based Brinkman term is proposed to penalize the velocity in solid area, which can expand the region controlled by the NS flow equilibrium equation from fluid to whole design domain, and can also establish the relationship between design variables and governing equations. Then, a MMV-based optimization model of NS flow is proposed and the design variables are updated by the method of moving asymptotes according to the sensitivity analysis. In benchmark examples, we can precisely track the topology boundary with detailed geometric information without the intuitive geometric judgement error. Also, by means of contrasting the results of proposed method with previous work, the effectiveness and advantages of the MMV method can be confirmed.

Periodic and almost periodic motions of Navier–Stokes flows and their stability in a perturbed half-space

Periodic and almost periodic motions of Navier–Stokes flows and their stability in a perturbed half-space

Monge–Ampère geometry and vortices

We introduce a new approach to Monge–Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge–Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier–Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge–Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge–Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier–Stokes equations, such as Hill’s spherical vortex and an integrable case of Arnol’d–Beltrami–Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.

Finite speed axially symmetric Navier-Stokes flows passing a cone

Let D be the exterior of a cone inside a ball, with its altitude angle at most π/6 in R3, which touches the x3 axis at the origin. For any initial value v0=v0,rer+v0,θeθ+v0,3e3 in a C2(D‾) class, which has the usual even-odd-odd symmetry in the x3 variable and has the partial smallness only in the swirl direction: |rv0,θ|≤1100, the axially symmetric Navier-Stokes equations (ASNS) with Navier-Hodge-Lions slip boundary condition have a finite-energy solution that stays bounded for all time. In particular, no finite-time blowup of the fluid velocity occurs. Compared with standard smallness assumptions on the initial velocity, no size restriction is made on the components v0,r and v0,3. In a broad sense, this result appears to solve 2/3 of the regularity problem of ASNS in such domains in the class of solutions with the above symmetry. Equivalently, this result is connected to the general open question which asks that if an absolute smallness of one component of the initial velocity implies the global smoothness, see e.g. page 873 in Chemin et al. (2017) [6]. Our result seems to give a positive answer in a special setting.As a byproduct, we also construct an unbounded solution of the forced Navier Stokes equation in a special cusp domain that has finite energy. The forcing term, with the scaling factor of −1, is in the standard regularity class, and it can be generated by an electric current in a long and straight wire (i.e. Ampère force). This result confirms the intuition that if the channel of a fluid is very thin, arbitrarily high speed in the classical sense can be attained under a mildly singular, physically reasonable force.

Memory-Efficient Interpolatory Projection Techniques for the Stabilization of Incompressible Navier-Stokes Flows

In this article, we are proposing an updated form of Krylov subspace-based interpolatory projection techniques for the stabilization of incompressible Navier-Stokes flows. In the proposed techniques, we utilize the reduced-order modelling approach implicitly, where reduced-order matrices need not be gathered explicitly. To estimate the aimed optimal feedback matrix, only the factored solution of the desired continuous-time algebraic Riccati equation (CARE) needs to be stored through the classical eigenvalue decomposition. The sparse structure of the target systems will remain invariant within the matrix-vector operations for generating the bases of the projector matrices through Krylov subspace techniques, where a cohesive projection scheme will be incorporated to ensure the potency of the projector matrices. So, the proposed techniques will be feasible for memory allocation and enhance the rapid convergence of the simulation. Analysis of the target systems’ transient characteristics, such as eigenvalues and step-responses, will be used to ascertain the competence and reliability of the proposed techniques. Necessary computation will be done numerically through MATLAB. Stabilization of the transient behaviors and minimization of the simulation time are the prime concerns in this work. Eventually, by comparing with the contemporary techniques the advancement of the proposed techniques will be confirmed.