Disturbance Equations Research Articles

Linear and nonlinear stability analysis of double-diffusion convection in an inclined Brinkman porous media with a concentration-based internal heat source

The onset of double-diffusion convection in an inclined porous medium with a concentration-based internal heat source is investigated by performing linear and nonlinear stability analysis. The Brinkman model is employed to model the momentum equation. Effects of different parameters, such as the thermal Rayleigh number (RaT) and solutal Rayleigh number (Ras), the angle of inclination (ϕ), the Lewis number (Le), the Darcy number (Da), and the concentration-based internal heat source (Q), are shown. A normal mode technique has been employed on the disturbance equations to get the generalized eigenvalues problem, which is solved by the Chebyshev-tau method coupled with the QZ algorithm in MATLAB. It was observed that increasing the solutal Rayleigh number stabilizes the system due to the higher concentration at the lower boundary than the upper boundary. It has also been observed that decreasing the Darcy number has a destabilizing effect, which means that decreasing permeability advances the onset of double-diffusion convection. Furthermore, it was observed that an increase in the concentration-based internal heat source destabilizes the system. Our numerical results show that for Ras>0 and ϕ>0°, for all Q values, the subcritical instability only exists for transverse rolls.

Solving the Small Disturbance Equation and Numerical Method Optimization in Subsonic and Supersonic Flows

This paper addresses the solution of the Small Disturbance Equation (SDE) under subsonic and supersonic flow conditions using Successive Over-Relaxation (SOR) techniques. The study implements and validates numerical methods tailored to the unique dynamics of these regimes by setting precise boundary conditions, adjusting computational stencils, and managing the Mach number ( M∞ ) across the solution domain. A detailed comparison of streamline patterns between the regimes illustrates the efficacy of the applied numerical strategies. In the subsonic domain, the flow demonstrates uniform and smooth characteristics, conducive to standard SOR techniques, as evidenced by the rapid decline in residual errors, confirming the method’s efficiency for accurate solutions. Conversely, the supersonic regime presents increased complexity where standard finite difference methods encounter notable challenges, necessitating more sophisticated approaches to capture the intricate flow behaviors effectively. Conversely, in the supersonic regime, where the flow behavior exhibits more complex characteristics, the standard finite difference method faces challenges.

Comparative Study on the Application Efficacy of Small Disturbance Equation versus One-Dimensional Euler Equation in Nozzle Flow Analysis

This paper presents a numerical simulation study of subsonic and supersonic nozzle flow regimes utilizing the Small Disturbance Equation (SDE), implemented through Python to analyze flow stability under various boundary conditions. The SDE, extensively applied in aerospace, meteorology, and fluid mechanics, offers a critical framework for examining aircraft stability and maneuverability, essential for ensuring flight safety. Additionally, its application extends to spacecraft stability and control in aerospace science. The findings from this study reveal that subsonic flows, characterized by their stability and smoothness, respond more predictably under varying boundary conditions compared to their supersonic counterparts. Conversely, supersonic flows demonstrate increased sensitivity to changes in boundary conditions, resulting in more complex flow patterns. This sensitivity underscores the need for precise control mechanisms in supersonic applications to maintain flow stability and ensure the safety and efficiency of aerospace operations. The simulations underscore the practical importance of the SDE in advancing the understanding of dynamic flow problems across different scientific and engineering disciplines.

Research and Analysis of the Small Disturbance Equation in Subsonic, Transonic, and Supersonic Regimes

This paper explores the application of the Small Disturbance Equation (SDE) across subsonic, transonic, and supersonic flow regimes. Derived from the Euler and Navier-Stokes equations, the SDE offers an efficient framework for analyzing aerodynamic behaviors, particularly through the utilization of discretization techniques and iterative solving methods executed in Python. The study assesses the accuracy and limitations of the SDE in detailing essential flow characteristics, revealing that while the equation performs effectively in subsonic and transonic flows, it encounters challenges in supersonic regimes. Nonlinear effects such as shock waves significantly hinder its performance at high speeds. Compared with conventional computational fluid dynamics (CFD) methods, the SDE stands out in scenarios where computational efficiency is paramount. However, its limitations in handling high-speed flows must be carefully considered, highlighting the need for further refinement in its application to supersonic dynamics. This analysis suggests that while the SDE is beneficial for certain aerodynamic studies, its scope and utility are constrained by the inherent complexities of high-speed fluid dynamics.

Non-Oberbeck–Boussinesq effects on the linear stability of a vertical natural convection boundary layer

The non-Oberbeck–Boussinesq effects on the stability of a vertical natural convection boundary layer are investigated using the linearised disturbance equations for air flows up to a temperature difference of $\Delta T=100\,{\rm K}$ . Based on the linear stability results, the neutral curve is shown to be sensitive to the choice of reference temperature. When evaluated using the film temperature $T_f$ , a lower film Grashof number is required to trigger the linear instability for larger $\Delta T$ . The relative contributions of shear and buoyant production to the perturbation kinetic energy budget reveals that the marginally unstable modes are amplified based on different mechanisms: for lower wavenumbers at relatively small Grashof number, the instability is driven by buoyancy; whereas for higher wavenumbers and larger Grashof number, the flow becomes unstable due to a shear instability. The use of reference temperature is found to scale the shear- and buoyant-driven instabilities differently so that no single reference temperature definition would collapse the neutral curves. The linear stability result further demonstrates that at a given Grashof number a higher temperature difference would give a larger amplification rate of the perturbation, which then leads to an earlier onset of the nonlinearities when evaluated at $T_f$ . Finally, by comparing the amplification rates obtained from direct numerical simulation and the linear stability results, the extent of the linear regime is determined for $\Delta T = 100\,{\rm K}$ .

Innovative Numerical Method to Partially Solve the Airfoil Flow

With the meteoric rise of modern computer science, computational fluid dynamics (CFD) has evolved into a formidable tool adept at addressing multifaceted challenges across diverse domains. Historically, CFD's maiden application was in the aviation sector, aimed at refining the aerodynamic contours of airfoils. Fast forward several decades and its deployment in this arena has reached unparalleled sophistication. This paper harnesses numerical methodologies within CFD to decipher flow dynamics around an airfoil. Central to this exploration is the small disturbance equation, a derivative of the full potential equation. The study meticulously discretizes this equation using advanced mathematical techniques. Furthermore, boundary conditions, namely the wall side and Neumann, are judiciously employed to ascertain boundary point values. Leveraging Python-based iterations, we derive numerical solutions for streamlining patterns and Mach number contours. A rigorous error analysis affirms the validity and efficacy of our results. While the study acknowledges certain inherent limitations, the findings compellingly delineate the flow characteristics around the airfoil in both subsonic and supersonic scenarios, offering valuable insights for future aerodynamic endeavors.

Research on 2D SDE Method of Fluid Field Effect above Airfoil at Transonic Speed

In this research, a Python-centric approach is proposed for simulating fluid dynamics over transonic airfoils. The study employs the finite difference method to solve the small disturbance equation (SDE), incorporating Newman boundary conditions. Iterative computations are adeptly managed using the successive over-relaxation (SOR) method. These methodologies, as validated through rigorous code testing, enable precise calculations of velocity distributions and vivid visualizations of fluid velocity streamlines. For this investigation, the airfoil is conceptualized as a minor obstruction and is abstracted into a trigonometric function. A pivotal aspect of this study is the exploration of the interplay between the Mach number and the 'shark angle'—a critical determinant in aerodynamic performance. Simulation outcomes reveal an inverse relationship between the Mach number and the airfoil's shark angle; a surge in the Mach number corresponds to a reduction in the shark angle. These findings, encapsulated in flow field diagrams, offer invaluable insights for the strategic design and optimization of airfoils, aiming to maximize lift while concurrently curtailing drag, thereby enhancing overall aerodynamic efficiency.

Various Flow in a Slender Tube: Mach Number and Domain Shape Influenced by Boundary Obstacles Via Small Disturbance Equations

With the exponential advancements in computer hardware and software in recent years, computational fluid dynamics (CFD) has emerged as a highly efficient alternative to traditional fluid mechanics studies. Its prowess in accurately predicting and analyzing flow mechanics has led to widespread adoption across diverse engineering disciplines. This research delves into a numerical analysis of flow conditions utilizing the small disturbance equation (SDE). The study bridges the gap between mathematical formulations and computational language, exemplified through Python code, by harnessing the successive over-relaxation (SOR) method, Neumann boundary conditions (BC), and the ghost point technique. The research underscores its predictions' precision in subsonic and supersonic scenarios by meticulously evaluating the resultant errors. This accuracy is further corroborated by examining overarching flow patterns and Mach number distributions. However, the program's outputs indicate a discernible lack of precision in the transonic case, with errors surpassing acceptable thresholds. This discrepancy is hypothesized to stem from potential mischaracterizations of points at the inflow and outflow boundaries. This study, thus, offers invaluable insights while also highlighting areas necessitating further refinement.

Effect of Magnetic Interaction Parameter and the Electrical Conductivity of the Fluid on the Stability of Hydromagnetic Channel Flow

The influence of magnetic interaction parameter and conductivity of the fluid on the stability against small perturbations on the streamlined base flow between two infinitely long parallel fixed plates is studied numerically. By normal mode analysis, the disturbance equations are reduced to Orr-Sommerfeld-type. Using the energy method, sufficient conditions for stability are derived by using the nature of the growth rate and sufficiently small values of the Reynold numbers. The disturbance equations are then solved using the Galerkin method corresponding to the base functions as Legendre-polynomials. Critical values for the Reynolds number, wave number, and speed of the wave are computed for various ranges of the magnetic interaction parameter and the magnetic Reynolds number. The curves of neutral stability are presented for different values of the nondimensional parameters that appeared in this study. The stability analysis is also discussed with the help of the plots of the rate of growth of disturbances for several values of the electrical conductivity and the magnetic interaction parameter. It is observed that both the fluid conductivity and the magnetic interaction parameter have direct control over fluctuations in the system. The results of this study are accurate and are comparable with the existing literature in the absence of a parallel magnetic field.

The Onset of Instability in A Magnetohydrodynamic Channel Flow through Porous Media of Casson Fluid

A detailed study is made on the stability of linear two-dimensional disturbances of Plane Poiseuille Flow (PPF) of Casson fluid through porous media in the presence of a vertical uniform magnetic field, B0 which is extremely useful in metals, mines, and fuels industries. Using the method of normal modes, the disturbance equations are derived. The resulting eigenvalue problem is then solved by the spectral collocation method using Chebyshev-based polynomials. The critical values of the triplets ( Rec, αc, cc ) are obtained for various values of the Casson parameter, η , Hartmann number, Ha , and porous parameter, σp. The stability of the system is discussed using the neutral stability curves for each value of the parameters present in the problem. It is found that the stability regions are enlarged for small values of η and large values of the porous parameter, σp and Hartmann number, Ha. It is also observed that the stability characteristics of plane Poiseuille flow in a porous medium are remarkably different from non-porous cases. The results obtained here contribute to the contemporary efforts to better understand the stability characteristics of PPF of Casson fluid flow through porous media in the presence of a uniform transverse magnetic field.

Influence of pressure disturbance wave on dynamic response characteristics of liquid film seal for multiphase pump

Slug flow or high GVF (Gas Volume Fraction) conditions can cause pressure disturbance waves and alternating loads at the boundary of mechanical seals for multiphase pumps, endangering the safety of multiphase pump units. The mechanical seal model is simplified by using periodic boundary conditions and numerical calculations are carried out based on the Zwart-Gerber-Belamri cavitation model. UDF (User Define Function) programs such as structural dynamics equations, alternating load equations, and pressure disturbance equations are embedded in numerical calculations, and the dynamic response characteristics of mechanical seal are studied using layered dynamic mesh technology. The results show that when the pressure disturbance occurs at the inlet, as the amplitude and period of the disturbance increase, the film thickness gradually decreases. And the fundamental reason for the hysteresis of the film thickness change is that the pressure in the high-pressure area cannot be restored in a timely manner. The maximum value of leakage and the minimum value of axial velocity are independent of the disturbance period and determined by the disturbance amplitude. The mutual interference between enhanced waves does not have a significant impact on the film thickness, while the front wave in the attenuated wave has a promoting effect on the subsequent film thickness changes, and the fluctuation of the liquid film cavitation rate and axial velocity under the attenuated wave condition deviates from the initial values. Compared with pressure disturbance conditions, alternating load conditions have a more significant impact on film thickness and leakage. During actual operation, it is necessary to avoid alternating load conditions in multiphase pump mechanical seals.

Airfoil design with computational fluid dynamics

In many industries, there is a need to model the flow of air over structural components. With sufficient information from these models, engineers can better implement these parts into a complete design. The purpose of this paper is to provide a model of specific airfoils using computational fluid dynamics (CFD). With computational fluid dynamics, the characteristics of air around an airfoil can be modeled, providing useful data to engineers who could be designing an airfoil or airplane. The CFD calculations are performed using Python, along with the two packages Numpy and Matplotlib. The governing equations of CFD, including Newton's Second Law, small disturbance equation (SDE), wave propagation, etc. are discretized and transformed into partial differentiation equations (PDE). Using the second order derivative of the wave propagation PDE, the SDE can be solved in iterations and plotted on a graph showing the velocity distributions for a particular airfoil. The results from the CFD calculations show general trends in velocity distributions, regardless of airfoil shape. These include a decrease in x-direction velocity at the ends of an airfoil with an increase at the midsection of the airfoil. Also, y-direction velocity is generally positive and increasing at the front of the airfoil, but negative and decreasing at the end of the airfoil. What is important to understand is how different airfoil shapes can change velocity distributions, moving to using 3D CFD calculations, and the possibility of using CFD for modeling airflow over a multitude of objects.[ Henry Bao, the first author, participated in the Illinois junior academy of science state fair, and abstracts of the regional winners' presentations were posted online. (ilacadofsci.com)]

Linear disturbance growth induced by viscous dissipation in Darcy–Bénard convection with throughflow

Modal and non-modal linear stability analyses are employed to investigate the effect of internal and external heating on disturbance temporal growth for the Darcy–Bénard convection with throughflow. A matrix-forming approach is employed for both purposes, where the generalised eigenvalue problem is built using the generalised integral transform technique. Although the disturbance equations are not self-adjoint, the non-modal analysis indicates that there is no transient growth. Hence, any disturbance growth in time must be induced by modal mechanisms. An absolute instability analysis reveals that viscous dissipation has a destabilising effect and introduces new modes that are eventually destabilised by increasing the Péclet number. Beyond critical values of the Péclet number, where codimension-two absolutely unstable points exist, these modes become more unstable than the classical mode found in the absence of viscous dissipation, which is stabilised by an increasing Péclet number. This internal heating mechanism generated by viscous dissipation is so strong at high enough Péclet numbers that instability becomes possible through heating from above.

Optimal Transient Energy Growth of Two-Dimensional Perturbation in a Magnetohydrodynamic Plane Poiseuille Flow of Casson Fluid

Abstract The study investigates the influence of the Casson fluid parameter and the spanwise uniform magnetic field on the onset of instability against infinitesimal disturbances in an electrically conducting fluid flow between two parallel nonconducting rigid plates. The fourth-order linearized disturbance equation governing stability is solved using the spectral collocation method with Chebyshev-based polynomials. The aim is to analyze in detail the effect of the parameters involved in the problem using both modal and nonmodal linear stability analysis. The modal analysis provides accurate values of the critical Reynolds number, critical wave number, and critical wave speed, denoted as critical triplets (Rc, αc, cc). Additionally, it examines the eigen-spectrum, growth rate curves, and neutral stability curves. On the other hand, the nonmodal analysis investigates the transient energy growth G(t) of two-dimensional (2D) optimal perturbations, the pseudospectrum of the non-normal Orr–Sommerfeld (O–S) operator (ℒ), and the regions of stability, instability, and potential instability of the fluid flow system. The extensive examination of both long-term behavior through modal analysis and short-term behavior through nonmodal analysis reveals that the Hartmann number (Ha) acts as a stabilizing agent, delaying the onset of instability. Conversely, the Casson parameter (η) acts as a destabilizing agent, advancing the onset of instability. The results obtained here are verified to be in good agreement with the existing literature in the absence of the Casson fluid flow parameter.

The study of the effect of fluid field above an airfoil under 2D small disturbance equation method

This review article presents a simulation based on Python for studying fluid flow over a small disturbance (airfoil) in subsonic and supersonic conditions. The simulation revolves around the small disturbance equation method, focusing on comparative analysis under supersonic conditions. Two schemes are applied in different conditions: the central difference scheme for subsonic flows and the upwind scheme for supersonic flows. Successive-over relaxation method is also utilized for converging iterative processes. The study focuses on one influence factor in aerodynamics, the Mach number, and investigates its impact on the shark angle of an airfoil. The simulation results demonstrate that the Mach number is inversely proportional to the shark angle of an airfoil, meaning that higher Mach numbers lead to smaller shark angles of the airfoil. Therefore, the Mach number can be considered a critical parameter in designing aerodynamic applications. This finding is consistent with prior research, indicating that the designed method is reliable.

Prediction of Aircraft Surface Noise in Supersonic Cruise State

The aerodynamic noise of an aircraft leads to vibration fatigue damage to structures. Herein, a prediction method for aircraft surface noise under the comprehensive effect of mixed acoustic sources during flight, primarily surface aerodynamic, air intake, and tail nozzle jet noises, was studied. In the supersonic cruising state, the internal and external flow fields of the aircraft were solved using the Reynolds-averaged Navier–Stokes equations to obtain the statistical average solution of the initial turbulence. The non-linear disturbance equation was used to obtain the surface acoustic load of the aircraft. The calculation results revealed that the main source of aircraft surface noise is aerodynamic noise. The sound pressure level on the fuselage increases gradually from front to rear along the aircraft, and the OASPL at the air intake and tail nozzle is relatively large. The jet noise has little effect on the sound pressure level at the front of the fuselage and only contributes to the OASPL at the tail nozzle of the fuselage. The intensity of pressure pulsations from the engine exhaust in the tail section is 93.3% of the total intensity of pressure pulsations.

SPECIFIC ACOUSTIC-GRAVITY WAVE MODES IN ISOTHERMAL ATMOSPHERE

In the paper, we show that the spectrum of acoustic-gravity waves in an isothermal atmosphere includes four specific evanescent modes. These modes are the solutions of the system of hydrodynamic equations for small atmospheric disturbances under the assumption that one of the quantities (horizontal or vertical components of particle velocity, density fluctuations, or temperature) is equal to zero. Three of the four specific modes (the Lamb wave, the Brunt-Väisälä oscillation, and the f-mode) are well known, but they were previously obtained as independent solutions. The recent discovery by the authors of the evanescent γ-mode made it possible to show that all four specified modes form a certain family of special modes of the isothermal atmosphere. On the spectral diagram of the frequency and the wave vector, there are four dispersion curves of these special modes in which one of the perturbed quantities is equal to zero. These curves belong to the evanescent region of the acoustic-gravity wave spectrum. They intersect each other at five points. It is shown that the specific modes cannot interact at the intersection points. The polarization ratios between two perturbed quantities have a different sign on either side of a particular curve if one of the quantities on this curve is zero. These properties can be used as indicators of the specific modes in experimental studies of the evanescent spectrum of AGWs. By using polarization relations, the possibility of observing these modes in the Earth’s atmosphere and on the Sun is also analyzed.

Density and Small Disturbance Equation-Based Simulation of Designed Bipalne Airfoil Under Supersonic Flow

For inviscid airflows, the small disturbance equation (SDE), as a condensed form of the full potential equation, is an effective approach for characterising the behaviour of this type of airflow in the presence of thin obstacles in its path. Density-based solutions (DBS) are frequently used by ANSYS to model the airflow behavior under the same environment. Through the impact of shock interference between the target wings, special Bussemann-type supersonic biplane designs can be effective in restricting sonic booms and wave drags in particular supersonic conditions. The modeling investigations of the shock oscillations produced between the wings are quite rare, nevertheless. Therefore, in order to replicate the behavior of the airflow over the target wing in supersonic settings and to compare the results more closely to reality, the Small Disturbance Equation (SDE) and the Density Based Solution (DBS) will be employed as efficient simulation tools in this study. The SDE results are vulnerable to inaccuracies because of the constraints of the simulation settings, whereas the DBS results are more accurate when compared to the exact simulation results. Overall, the DBS simulation results and the legit results are consistent, indicating that the inviscid airflow hypothesis is a feasible option for modelling the effectiveness of shock oscillations between wings. The SDE approach, which is implemented in Python and is a condensed version of the en-tire potential equation, offers a lot of space for development as it only does simulations in terms of velocity in this article and ignores the pressure and density elements.

The Influence of Fluid Dynamics on the Design of Different Vehicle Structures

It is of great significance to understand the airflow disturbance above the vehicle for the design of the vehicle model. However, the traditional method is wind tunnel test, which is inconvenient and time wasted. In this article the team simulated the curves of different shapes above cars, such as Beetle, modern sedan car, jeep, and wedge-shaped cars. Besides, calculated the small disturbance equation in the form of python code. This paper anaylised the full potential equation to find the small disturbance equation. Then studied and analyzed the disturbance of subsonic airflow above cars. Finally, this paper analysed the functional image and did a conclusion. The results shows that different shape of cars has different flow above the car in the subsonic flow and the wedge car has almost the best shape. The experiment result shows that it has an exelent accuracy compared to the literature, besided it has a low cost and a shorter time waste.

The impact of inlet mach number and domain shape on subsonic nozzle flow via small disturbance equation

Computational fluid dynamics (CFD) is a cross-disciplinary field that combines mathematics, fluid dynamics, and computer programming to simulate and analyze fluid flow problems using numerical methods. Nowadays, the use of CFD to conduct numerical simulations rather than costly and time-consuming ignition experiments has become an essential aspect of nozzle design. Therefore, it is significant to study the nozzle by numerical simulation. In this paper, a numerical method called small disturbance equation (SDE) is adopted to investigate the impact of inlet Mach number and domain shape on subsonic nozzle flow. Conducting the small disturbance equation via python code, the impact of the inlet Mach number and domain shape is revealed by comparing the different errors of the method, streamlines of the nozzle flow, velocity fields. It is found that with the increase of inlet Mach number, the error of the SDE method grows, and the velocity in x-direction increases while the velocity in y-direction stays the same. Besides, with the shrink of the domain shape, the error of the SDE method gets larger, meanwhile, the maximum of the velocity in each direction increases while the minimum of the velocity decreases. This research would make advantages to identify the optimal designs of a subsonic nozzle.