Dimensional Navier Research Articles

Application of Triple- and Quadruple-Generalized Laplace Transform to (2+1)- and (3+1)-Dimensional Time-Fractional Navier–Stokes Equation

In this study, the solution of the (2+1)- and (3+1)-dimensional system of the time-fractional Navier–Stokes equations is gained by utilizing the triple-generalized Laplace transform decomposition method (TGLTDM) and quadruple-generalized Laplace transform decomposition method (FGLTDM). In addition, the results of the offered methods match with the exact solutions of the problems, which proves that, as the terms of the series increase, the approximate solutions are closer to the exact solutions of each problem. To verify the appropriateness of these methods, some examples are offered. The TGLTDM and FGLTDM results indicate that the suggested methods have higher evaluation convergence as compared to the ADM and HPM.

Error estimates for finite element discretizations of the instationary Navier–Stokes equations

In this work we consider the two dimensional instationary Navier–Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the L∞(I; L2(Ω)), L2(I; H1(Ω)) and L2(I; L2(Ω)) norms have been shown. The main result of the present work extends the error estimate in the L∞(I; L2(Ω)) norm to the Navier–Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the L∞(I; L2(Ω)) error estimate, also allow us to show best approximation type error estimates in the L2(I; H1(Ω)) and L2(I; L2(Ω)) norms, which complement this work.

Thermochemical Exploration of a Cavity Based Supersonic Combustor with Liquid Kerosene Fuel

Thermochemical exploration of a liquid hydrocarbon fueled scram jet combustor is presented. Three dimensional Navier Stokes equations along with K-ε turbulence model and single step kerosene-air reaction kinetics are solved using commercial software. Various combustor configurations with different fuel injection cavities are analyzed. Simulations capture all the essential features of the flow field. Good comparisons between computational and experimental surface pressure form the basis for further analysis. Parametric studies have been carried out with different droplet diameters to study its effect in the flow development. The numerical simulation also confirmed the experimental observation that the threshold value of length-to-depth ratio for cavity characterization is different for reacting and non-reacting flows.

Numerical Investigation of Confinement Effect on Supersonic Turbulent Flow Past Backward Facing Step With And Without Transverse Injection

Effect of confinement has been investigated numerically for supersonic turbulent flow past backward facing step in a nonreacting scramjet combustor. Flow structure downstream of the backward facing step has been studied by considering various configurations with different combustor heights as well as unconfined flow. Staged transverse sonic injectors with different combustor heights are also considered to find out the effect of confinement on the penetration, spreading and other flow features in the flow field. Three dimensional Navier Stokes equations along with k-ε turbulence model are solved using a commercial CFD software. The simulation captures all essential features of the flow. Good comparisons of various flow profiles have been obtained between experimental and computed values. Although confinement creates complicated shock reflections in the combustor, the length of the recirculation bubble behind the backward facing step remains almost constant. For the injection case, the recirculation region is extended upto the bow shock arising upstream of the injector and two recirculation bubble is seen between the backward facing step and the injector.

CFD Prediction of Ramjet Intake Characteristics at Angle of Attack

The characteristics of installed intakes of a ramjet engine at angle of attack are predicted numerically. Three dimensional Navier Stokes equations are solved alongwith k - ε turbulence model using commercial CFD software. The software was first validated for isolated intake and computed results match very well with theoretical results available in literature. Pressure recovery Vs. mass flow characteristics of four installed intakes placed in a rear location of a ramjet vehicle is evaluated at different angles of attack upto 6°. Intake performance matches reasonably well with experimental results. It has been observed that significant variation of intake performance exists between windward and leeward intakes. For higher angle of attack, the leeward intake move towards the subcritical operation faster and the intake flowfield is seen to interact with the corebody boundary layer causing significant spillage.

Numerical Simulation of Underexpanded Sonic Jet

Underexpanded sonic jet is numerically simulated using commercial CFD software. Three dimensional Navier Stokes equations alongwith two equation turbulence models are solved. Computed centerline distribution and profiles of various flow variables at various axial stations agree quantitatively with experimental results. The comparison of the flow variables computed with four different two equation models reveals that k - e and RNG k - e models perform very well in predicting both pressure and temperature throughout the flow filed; whereas SST and k - w models fail to predict the flow behavior beyond the first Mach disc.

Time-fractional (2+1)-dimensional navier-stokes equations: similarity reduction and exact solutions for one-parameter lie group of rotations

The current study is dedicated to solving the time-fractional (2+1)-dimensional Navier–Stokes model. The model has wide applications in blood flow, in the design of power stations, weather prediction, ocean currents, water flow in a pipe, air flow around the aircraft wings, the analysis of pollution, and many other areas of engineering. The Lie symmetry approach is applied to the governed time-fractional equation to fulfill this need. In the direction of exact solutions of the time-fractional equation first of all invariance condition is obtained in the presence of the Lie group. Consequently, infinitesimals are obtained with the help of the invariant condition. Moreover, these infinitesimals are utilized to obtain the subalgebras. Further, under each subalgebras similarity variables and similarity solutions are obtained which are used to find the reduced equations. These reduced equations are solved for exact solutions. The solutions of the reduced equations are further used to find the exact solutions of the main time-fractional (2+1)-dimensional Navier–Stokes equation with the help of similarity solutions under each subalgebra.

A variational approach to hyperbolic evolutions and fluid-structure interactions

We show the existence of a weak solution for a system of partial differential equations describing the motion of a flexible solid inside a fluid: A nonlinear, viscoelastic, n -dimensional bulk solid governed by a PDE including inertia is interacting with an incompressible fluid governed by the ( n -dimensional) Navier–Stokes equation for n\geq 2 . The result is the first allowing for large bulk deformations in the regime of long time existence for fluid-structure interactions. The existence is achieved by introducing a novel variational scheme involving two time-scales that allows us to extend the method of minimizing movements to hyperbolic problems involving nonconvex and degenerate energies.

Uniform Linear Inviscid Damping and Enhanced Dissipation Near Monotonic Shear Flows in High Reynolds Number Regime (I): The Whole Space Case

We study the dynamics of the two dimensional Navier Stokes equations linearized around a strictly monotonic shear flow on $${\mathbb {T}}\times {\mathbb {R}}$$ . The main task is to understand the associated Rayleigh and Orr–Sommerfeld equations, under the natural assumption that the linearized operator around the monotonic shear flow in the inviscid case has no discrete eigenvalues. We obtain precise control of solutions to the Orr–Sommerfeld equations in the high Reynolds number limit, using the perspective that the nonlocal term can be viewed as a compact perturbation with respect to the main part that includes the small diffusion term. As a corollary, we give a detailed description of the linearized flow in Gevrey spaces (linear inviscid damping) that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The key difficulty is to accurately capture the behavior of the solution to Orr–Sommerfeld equations in the critical layer. In this paper we consider the case of shear flows on $${\mathbb {T}}\times {\mathbb {R}}$$ . The case of bounded channels poses significant additional difficulties, due to the presence of boundary layers, and will be addressed elsewhere.

Topological asymptotic expansion for the full Navier–Stokes equations

This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.

The Cauchy Problem for the \(\boldsymbol{N}\)-Dimensional Compressible Navier–Stokes Equations without Heat Conductivity

The Cauchy Problem for the \(\boldsymbol{N}\)-Dimensional Compressible Navier–Stokes Equations without Heat Conductivity

Short engine intakes: Design and trade-off aerodynamic recommendations

This work sets forth a design procedure for compact, highly diffusive engine intakes for high subsonic speed applications. The aerodynamic tradeoffs between cruise and take-off flights are discussed, as well as methods to enhance take-off performance without negatively impacting high-speed cruise performance. Intake performance is integrated into overall engine analysis to help guide future mission analyses. A multi-objective optimization routine is performed for compact engine intakes at a Mach number of 0.9 using two dimensional axisymmetric Reynolds Averaged Navier Stokes simulations. This optimization routine yielded a family of related curves that maximize intake diffusive capability and minimize pressure losses. Design recommendations to achieve intakes with minimal pressure losses and maximal diffusion, including intake length and inlet lip design are discussed. Due to the high diffusion rate of the intake, the intake performance at take-off suffers greatly as measured by massflow ingestion. Methods to enhance take-off performance, from designing a variable geometry intake to creating slots to sliding intake components, are evaluated and ranked for future designers to get an order of magnitude understanding of the types of massflow enhancements. The slotted design was assessed through three-dimensional computations, and performance was compared to the isentropic limit. Additionally, the off-design performance of the intake is considered: with different Mach number flights, non-axial flow conditions, various altitudes, and unsteady engine operation considered. These off-design effects are evaluated to generate an intake map across a comprehensive engine operational envelope. Off-design intake performance is integrated into an engine model for mission planning based on intake performance. Finally, an experimental model of an optimized intake geometry is tested to validate the proposed numerical method.

Laminar Flow in a PEM Fuel Cell Cathode Channel

A Berman’s model for the laminar flow of incompressible fluid in a channel with permeable walls is extended for variable along the channel velocity of injection. The system of two–dimensional continuity and Navier–Stokes equations is reduced to a single ODE for the transversal velocity with coefficients depending on position along the channel. Numerical solution for the flow in the cathode channel of a PEM fuel cell is presented. The flow velocity profile across the channel is almost indistinguishable from the Poiseuille’s parabolic shape; however, the distribution of pressure gradient and longitudinal velocity differ quite significantly from the Berman’s result.

Conjugate MHD natural convection of hybrid nanofluids in a square enclosure containing a complex conductive cylinder

PurposeThe purpose of this study is to investigate the enhancement and suppression of heat transfer for hybrid nanofluids (Cu–Al2O3/water) in a square enclosure containing a thermal-conductive cylinder when the Lorentz force is applied to the hybrid nanofluids.Design/methodology/approachSince the inner conductive cylinder in present research has a complex geometry, an in-house meshless method, namely, the local radial basis function (LRBF) method, is applied to solve the 2 dimensional (2D) incompressible Navier–Stokes equation in the fluid domain and Fourier heat conduction equation in solid domain. The solid–fluid interface remains the physical continuity of temperature and heat flux. Only the Lorentz force is considered for the presence of the magnetic field. The conjugate natural convection is assumed to be steady, thus only fully developed heat exchange from the nanofluids to solid or vice versa is comprehensively investigated.FindingsIt can be concluded that Lorentz force plays a more significant role than hybrid nanofluids in enhancing/suppressing heat transfer when the orientation of magnetic field is the same to the x direction. The thermal conductivity ratio can dramatically change the isotherms and streamlines as well as the mean value of the Nusselt number, resulting in totally different heat transfer phenomena. The included angle of magnetic field also has a significant effect on the heat transfer rate when it changes from horizontal to vertical.Research limitations/implicationsThe constant thermo-physical properties of incompressible fluid and the 2D steady flow are considered in this study.Originality/valueThe conjugate MHD natural convection of hybrid nanofluids is numerically investigated by an in-house meshless LRBF method. The enhancement and suppression of heat transfer under the combined influence of the volume fraction of nanoparticles, Hartmann number and the thermal conductivity ratio are comprehensively investigated.

A comprehensive study of the onset of boundary layer separation in the unbounded flow around a circular cylinder

The onset of boundary layer separation in the unbounded flow around a circular cylinder is determined directly with a numerical experiment with accuracy to the second decimal point of the Reynolds number, Re. The governing equations are the steady state, two dimensional Navier–Stokes equations, which are solved with Galerkin finite elements. A new criterion, the entrance ratio E=D/L with D the diameter of the cylinder and L the entrance length of the domain, is introduced in addition to the traditional blockage ratio; the aim is to establish conditions for unbounded flow in both flow directions. A hypothesis is formulated and verified for the onset of separation leading to the conclusion that this flow undergoes a supercritical pitchfork bifurcation with respect to fixed points on the boundary. Results are presented for various blockage ratios, B, in the range 1.2238·10−8≤B≤0.005 and constant entrance ratio E=2.4472·10−9 until asymptotic behavior is observed for the onset of separation in the Reynolds number, in the angle and the length of the smallest eddy. Onset of separation occurs in the region 6.36≤Re<6.37. The bifurcation diagram of the flow is calculated. Streamlines of the flow domain before and after the bifurcation point are shown, including images of the smallest possible eddies. In addition, coefficients of base suction and total drag are computed and shown in the regime 0.1≤Re≤40. The results are compared with available laboratory and numerical measurements.

How to Extract a Spectrum from Hydrodynamic Equations

The practical results gained from statistical theories of turbulence usually appear in the form of an inertial range energy spectrum $${\mathcal {E}}(k)\sim k^{-q}$$ and a cutoff wavenumber $$k_{c}$$ . For example, the values $$q=5/3$$ and $$\ell k_{c}\sim \mathrm {Re}^{3/4}$$ are intimately associated with Kolmogorov’s 1941 theory. To extract such spectral information from the Navier–Stokes equations, Doering and Gibbon (Phys. D 165, 163–175, 2020) introduced the idea of forming a set of dynamic wavenumbers $$\kappa _n(t)$$ from ratios of norms of solutions. The time averages of the $$\kappa _n(t)$$ can be interpreted as the 2nth moments of the energy spectrum. They found that $$1 < q \leqslant 8/3$$ , thereby confirming the earlier work of Sulem and Frisch (J. Fluid Mech. 72, 417–423, 1975) who showed that when spatial intermittency is included, no inertial range can exist in the limit of vanishing viscosity unless $$q \leqslant 8/3$$ . Since the $$\kappa _n(t)$$ are based on Navier–Stokes weak solutions, this approach connects empirical predictions of the energy spectrum with the mathematical analysis of the Navier–Stokes equations. This method is developed to show how it can be applied to many hydrodynamic models such as the two dimensional Navier–Stokes equations (in both the direct- and inverse-cascade regimes), the forced Burgers equation and shell models.

The elliptical vortices, integrable Ermakov structure, Schrödinger connection, and Lax pair in the compressible Navier–Stokes equation

AbstractIn this paper, we investigate the (2+1)‐dimensional compressible Navier–Stokes equation with density‐dependent viscosity coefficients. We introduce a novel power‐type elliptic vortex ansatz and thereby obtain a finite‐dimensional nonlinear dynamical system. The latter is shown to not only have an underlying integrable Ermakov structure of Hamiltonian type, but also admit a Lax pair formulation and associated stationary nonlinear Schrödinger connection. In addition, we construct a class of elliptical vortex solutions termed pulsrodons corresponding to pulsating elliptic warm core eddies and discuss their dynamical behaviors. These solutions have recently found applications in geography, and oceanic and atmospheric dynamics.

Heat Transfer Enhancement of Impingement Cooling by Different Crossflow Diverters

Abstract Impingement cooling can effectively disperse local heat load, but its downstream heat transfer is always reduced due to crossflow effect. In this study, the flow and heat transfer characteristics of impingement cooling with semi-circular (SC), semi-rectangular (SR), semi-diamond (SD), and semi-four-pointed star (SFS) crossflow diverters are compared over the ReD ranging from 3,500 to 14,000 by solving three dimensional Reynolds-averaged Navier–Stokes equations with SST k–ω turbulence model. It is found that the arrangement of crossflow diverters changes the distribution of local jet Reynolds number (ReD,j/ReD) and reduces the mass velocity ratio of downstream crossflow to jet (Gcf/Gj), so the impingement heat transfer is enhanced significantly. However, friction loss also increases. Overall evaluation reveals that all crossflow diverters can improve the comprehensive heat transfer performance parameter (Φ), and the maximum increases of Φ are 11.0%, 14.3%, 12.2%, and 14.7% for SC, SR, SD, and SFS cases, respectively. It is noted that the Nusselt number of heated SFS-shaped diverter surface is also the highest. Besides, the influences of streamwise location (L) and thickness (t) of SFS-shaped diverter are also investigated. Results show that when the L increases from 2D to 3D, the heat transfer and friction loss change slightly; when the L increases from 3D to 4D, the heat transfer decreases sharply, and friction loss increases seriously. As for the t, it has almost no effect on the flow field and heat transfer.

Assessment of RANS-based turbulence model for forced plume dynamics in a linearly stratified environment

We present a computational study on the dynamics of a forced plume released in a linearly stratified medium for a range of buoyancy frequencies N∞=0 - 0.7 s−1. Three dimensional unsteady Reynolds-Averaged Navier–Stokes (RANS) simulations are performed using the standard k−ε turbulence model. The mean flow parameters such as the maximum height Zm, mean centerline velocity 〈wc〉, mean axial velocity 〈w〉, and turbulence parameters such as shear production (P), buoyancy production (B), and dissipation rate (ε) are compared with the experimental results of Mirajkar et al. (2020) and a good agreement is found. The validated model is then used to study the forced plume characteristics and energetics at various values of N∞. The passive scalar contours reveal that the plume reaches a maximum height in a stratified medium where the momentum becomes zero, then falls back to the neutral buoyancy height before spreading in the lateral direction. We also found that the maximum height reached by the plume decreases with increasing value of N∞, consistent with the previous studies. Quantification of turbulence parameters reveal that the production flux, P, and viscous dissipation, ε, from the RANS computation are in reasonable agreement with the experimental values at N∞ = 0.4 s−1. Further, the residual, given by 〈〈P〉〉 + 〈〈B〉〉 - 〈〈ε〉〉, is found to be larger for higher N∞, indicating more non-homogeneity in the flow at higher stratification. Overall, the RANS modeling is found to accurately predict the mean flow quantities, such as mean centerline velocity and maximum height reached by the plume. The comparison of the modeled turbulence quantities with experimental data seems satisfactory, but indicates a need for some improvement at high values of N∞.

Rotating tip clearance asymmetry and role of endwall injection in stability desensitization of a transonic fan

The current study is aimed at understanding the effect of rotating tip clearance asymmetry on the operability and performance of a transonic compressor. Another objective of this investigation is to determine the influence of tip injection on reducing the detrimental effects of clearance asymmetry. Three dimensional unsteady Reynolds-averaged Navier–stokes simulations have been performed from choke to stall for different arrangements of non-uniform blade heights in a transonic fan. Furthermore, numerical computations have been conducted with endwall injection of air. The numerical results have been validated against experimental data. Results show that having the same mean tip clearance, the asymmetric compressor is less stable than the axisymmetric configuration. However, the peak pressure rise is found to be almost linearly correlated to the mean tip clearance for both the axisymmetric and asymmetric compressors. It is found that tip injection can desensitize the compressor to the tip clearance asymmetry. Results further reveal that tip clearance asymmetry does not change the compressor path to instability. However, endwall injection is found to be able to change the compressor stalling mode. Investigations concerning rotating non-uniformity (caused by non-uniform blade heights) are very few in open literature. The obtained results can assist in predicting the effect of rotating tip clearance asymmetry on the stability and performance of high-speed compressor rotors. Furthermore, the results uncover how tip injection can desensitize the compressor stability and affect its path into instability, which is one of the most important questions in the turbomachinery world.