Origins of one dimensional instability in stationary shock and slowly moving shock

Shock instabilities in the numerical sense include the carbuncle phenomenon and the slowly moving shocks. The carbuncle phenomenon is a term referred to the protruding formation at the stagnation region in addition to the continuous bow shock when simulating a high-speed flow over a blunt body. Most schemes formulated to cure this problem only focus on the dissipation methods without properly indulged into the real cause, which could also be the root problem for the slowly moving shock. Therefore, this paper attempted to find the source of the problem by firstly analyzing the governing equations starting from 1D case. After using perturbation mechanism on the conservative variables, several factors were found and one of them is caused by perturbation in density. Then, a dissipation was added to the RHS (right-hand side) of the continuity equation to remove the perturbation. This artificial dissipation has shown stable solutions for both stationay and slowly moving shock problems

Analytical And Numerical Study On Carbuncle Phenomenon

Most newly developed schemes in the literatures to solve the shock instability in hyperbolic conservation laws mainly focused on adding ad hoc diffusion factor without properly indulging into the sources of the problem. An example of shock instabilities is the carbuncle phenomenon which occurs when simulating a blunt body subjected to a high speed flow. The shock formed ahead of the body is unphysical. Therefore, the goals of this study are to find at least one possible cause of the problem and to fix the instability from that cause. Extruding a possible source of the problem, herein the elimination process was applied to reduce the number of conservative variables involve, starting from the Burgers’ equation followed by isothermal equations to the full Euler equations. Then, a small perturbation definition to the hyperbolic conservation equations was used as a mean to ease the nonlinearity from the equations. After that, the method of normal mode was used to analytically analyze the instability mechanism. The cause was found to be the perturbation from density which seeding into the instability. Numerical tests were then used to check the validity of the analytical result and they gave a good agreement with the analysis. Finally, a tunable dissipative coefficient was inserted only to the density equation and a range value of 0:0

Thermal Marangoni flow past a permeable stretching/shrinking sheet in a hybrid Cu-Al2O3/water nanofluid

The present study accentuates the Marangoni convection flow and heat transfer characteristics of a hybrid Cu-Al2O3/water nanofluid past a stretching/shrinking sheet. The presence of surface tension due to an imposed temperature gradient at the wall surface induces the thermal Marangoni convection. A suitable transformation is employed to convert the boundary layer flow and energy equations into a nonlinear set of ordinary (similarity) differential equations. The bvp4c solver in MATLAB software is utilized to solve the transformed system. The change in velocity and temperature, as well as the Nusselt number with the accretion of the dimensionless Marangoni, nanoparticles volume fraction and suction parameters, are discussed and manifested in the graph forms. The presence of two solutions for both stretching and shrinking flow cases are noticeable with the imposition of wall mass suction parameter. The adoption of stability analysis proves that the first solution is the real solution. Meanwhile, the heat transfer rate significantly augments with an upsurge of the Cu volume fraction (shrinking flow case) and Marangoni parameter (stretching flow case). Both Marangoni and Cu volume fraction parameters also can decelerate the boundary layer separation process

Thermally Stratified Flow Of Cu-Al2O3/Water Hybrid Nanofluid Past A Permeable Stretching/Shrinking Circular Cylinder

The present study emphasizes the thermally stratified hybrid nanofluid flow due to a permeable stretching/shrinking cylinder. Thermal buoyancy force is also taken into consideration to incorporate with the thermal stratification process. An improved hybrid nanofluid (dual nanoparticles) may offer a better heat transfer performance in many engineering applications. In the present work, the combination of copper (Cu) and alumina (Al2O3) nanoparticles with water as the working fluid is analytically modeled using the extended form of Tiwari and Das nanofluid model. A suitable transformation is adopted to simplify the boundary layer and energy equations into a nonlinear system of ODEs. A boundary value problem solver with fourth order accuracy (bvp4c) in the MATLAB software is utilized to solve the transformed system. The change in velocity and temperature as well as the heat transfer rate and skin friction coefficient are deliberated and graphically manifested for appropriate values of the dimensionless stretching/shrinking, nanoparticles volume fraction, and thermal stratification parameters. The presence of dual solutions is seen on all the profiles within the range of selected parameters

Stagnation Point Flow Of Hybrid Nanofluid Over A Permeable Vertical Stretching/Shrinking Cylinder With Thermal Stratification Effect

Hybrid nanofluid is invented to improve the heat transfer performance of traditional working fluids (water, traditional nanofluid) in many engineering applications. The present study highlights the numerical solutions and stability analysis of stagnation point flow using hybrid nanofluid over a permeable stretching/shrinking cylinder. The combination of copper (Cu) and alumina (Al2O3) nanoparticles with water as the base fluid is analytically modeled using the single phase model and modified thermophysical properties. A set of transformation is adopted to reduce the complexity of the governing model and then, numerically computed using the bvp4c solver in Matlab software. Suction parameter is vital to generate dual similarity solutions in shrinking cylinder case while no solution is found if the surface is impermeable. Two solutions are possible for the assisting and opposing flow within a specific value of the buoyancy parameter. For the shrinking cylinder, Al2O3-water nanofluid has the lowest heat transfer rate than Cu-water and hybrid Cu-Al2O3/water nanofluids. A suitable combination of alumina and copper nanoparticles volumetric concentration in hybrid nanofluid can produce higher heat transfer rate than the Cu-water nanofluid. The execution of stability analysis reveals that the first solution is more realistic than second solution. However, the present results are only fixed to the combination of copper and alumina nanoparticles only and the other kind of hybrid nanofluid may have different outcome

Thermal Marangoni flow past a permeable stretching/shrinking sheet in a hybrid Cu-Al2O3/water nanofluid

The present study accentuates the Marangoni convection flow and heat transfer characteristics of a hybrid Cu-Al2O3/water nanofluid past a stretching/shrinking sheet. The presence of surface tension due to an imposed temperature gradient at the wall surface induces the thermal Marangoni convection. A suitable transformation is employed to convert the boundary layer flow and energy equations into a nonlinear set of ordinary (similarity) differential equations. The bvp4c solver in MATLAB software is utilized to solve the transformed system. The change in velocity and temperature, as well as the Nusselt number with the accretion of the dimensionless Marangoni, nanoparticles volume fraction and suction parameters, are discussed and manifested in the graph forms. The presence of two solutions for both stretching and shrinking flow cases are noticeable with the imposition of wall mass suction parameter. The adoption of stability analysis proves that the first solution is the real solution. Meanwhile, the heat transfer rate significantly augments with an upsurge of the Cu volume fraction (shrinking flow case) and Marangoni parameter (stretching flow case). Both Marangoni and Cu volume fraction parameters also can decelerate the boundary layer separation process

MHD Mixed Convective Stagnation Point Flow With Heat Generation Past A Shrinking Sheet

This paper investigates the in uence of magnetohydrodynamics (MHD) mixed convective stagnation point flow over a shrinking sheet with the enhancement of heat generation/source. Using appropriate similarity transformations, the model are transformed into a system of nonlinear equations and then solved using bvp4c built-in-function in Matlab. Numerical results are presented graphically for the distributions of velocity, temperature as well as the skin friction coefficient and local Nusselt number. The findings revealed the dual solutions obtained within a particular range of the mixed convection parameter and shrinking parameter. It is found that the fluid velocity increases with the increasing values of the magnetic and mixed convection parameter while opposite results obtained for the fluid temperature. A stability analysis was performed and it is proven that the first solution is physically realizable and stable whereas the second solution is unstable

Mixed Convective Stagnation Point Flow Of A Thermally Stratified Hybrid CU-AL2O3/Water Nanofluid Over A Permeable Stretching/Shrinking Sheet

The study scrutinizes the coupled effects of thermal stratification and mixed convection on boundary layer flow and heat transfer of a hybrid Cu-Al2O3/water nanofluid. Stretching/shrinking surface is permeable to allow the wall fluid suction while thermal convection is also included to deal with the thermal stratification phenomenon. In the present work, the combination of copper (Cu) nanoparticles and Al2O3/water nanofluid is modelled using the analytical hybrid nanofluid model. A similarity transformation is adopted to reduce the governing model into a set of ordinary (similarity) differential equations. The efficient boundary value problem with fourth order accuracy (bvp4c) solver in MATLAB software is utilized to solve the transformed model. An astonishing result is obtained where the heat transfer rate of hybrid nanofluid intensifies when small suction parameter is imposed on the stretching/shrinking sheet while a contrary result is obtained when higher value of suction is applied. Suction and opposing buoyancy parameters are among the control parameters that induce the existence of second solution. Stability analysis affirms that the first solution is mathematically stable. The present results are conclusive to the combination of alumina and copper nanoparticles only and other combination of nanoparticles may produce different flow and heat transfer characteristics

Magnetohydrodynamics (MHD) Flow And Heat Transfer Of A Doubly Stratified Nanofluid Using Cattaneo-Christov Model

The present study utilized Cattaneo-Christov heat flux model for solving nanofluid flow and heat transfer towards a vertical stretching sheet with the presence of magnetic field and double stratification. Thermal and solutal buoyancy forces are also examined to deal with the double stratification effects. Buongiorno’s model of nanofluid is used to incorporate the effects of Brownian motion and thermophoresis. The boundary layer with non-Fourier energy equations are reduced into a system of nonlinear ordinary (similarity) differential equations using suitable transformations and then numerically solved using bvp4c solver in MATLAB software. The local Nusselt and Sherwood numbers of few limited cases are tabulated and compared with the earlier published works. It showed that a positive agreement was found with the previous study and thus, validated the present method. Numerical solutions are graphically demonstrated for several parameters namely magnetic, thermal relaxation, stratifications (thermal and solutal), thermophoresis and Brownian motion on the velocity, temperature and nanoparticles volume fraction profiles. An upsurge of the heat transfer rate was observed with the imposition of the thermal relaxation parameter (Cattaneo-Christov model) whereas the accretion of thermal and solutal stratification parameters reduced the temperature and nanoparticles concentration profiles, respectively

Flow And Heat Transfer Past A Permeable Power-Law Deformable Plate With Orthogonal Shear In A Hybrid Nanofluid

This study concerns the three-dimensional hybrid nanofluid flow and heat transfer due to a deformable (stretching/shrinking) plate with power-law velocity and orthogonal surface shear. The flow due to the shrinking sheet is maintained with the imposition of wall mass suction. The effect of adding Cu and Al2O3 nanoparticles are represented by a homogeneous mixture model with the modified thermophysical properties. Two types of thermophysical properties for hybrid nanofluids are discussed and compared in this interesting work. The three-dimensional model is then, reduced into a relevant set of ordinary differential equations using similarity transformation. The results are generated using the bvp4c solver and presented in the tables and graphs. Duality of solutions are observed in both stretching and shrinking regions, however, only the first solution is proved to be stable and realistic. Surprisingly, the heat transfer rate augments when the power law velocity is used. The hybrid nanofluid with an upsurge of copper volume fraction also reduces the rate of heat transfe