Steady-state Momentum Equations Research Articles

The aerodynamic analysis of iced and clean NACA 4410 and 0012 airfoils

AbstractThe aims of the work is to develop a numerical iced airfoil; quarter round forward ice was tested on NACA 4410 and NACA 0012 airfoils at zero and non-zero angles of attack, and Reynolds numbers equal to (2∙105, 2.4∙105) based on airfoil chord and Mach number 0.04. The two-dimensional steady-state momentum equations with the continuity equation have been solved applying a finite volume method to examine the turbulent flow over a clean and iced airfoil. The cambered airfoil NACA 4410 spends less power than the unsymmetrical airfoil at the same angle of attack. The reported numerical results demonstrated that for airfoil NACA 4410, the drag was increased by 40%, and the lift was reduced by 22%. However, for airfoil NACA 0012, the drag was increased by 43%, and the lift was decreased by 21%.

Numerical 3D simulation of developing turbulent stratified gas-liquid flow in curved pipes consisting of entrained particles through this type of flow

Predicting multiphase flows in curved pipes is of great importance in industrial equipment. In the present study, a computational model for predicting the velocity profile is developed and used to study the developing turbulent gas-liquid- solid three dimension flow in curved pipes. In order to discretize and solve the three-dimensional steady-state momentum equations, the finite volume scheme on staggered grids besides central difference and QUICK scheme have been used. Moreover, the k-e model is employed to reflect the nature of turbulence in the flow. In order to address the needs for sooner convergence and convenient mapping of the physical domain, the computations have been performed in an extended toroidal coordinate system. Particle tracking has been done using Lagrangian approach in which two-way coupling regime is considered. In terms of validation, the numerical simulation results for the straight duct (infinite curvature), have been compared with the analytical solution and previous experimental results. Moreover, injection of particles through the flow indicates that, in each section of the bend, trade-off between centrifugal and pressure gradient forces plays a key role on particles motion. In last section, the effects of particle diameter and bend curvature on particle motion have been examined.

Reversal of Secondary Flows in a Sharp Channel Bend

Field observations of the currents at a bending channel in the Abashiri River in Hokkaido, Japan have revealed an unexpected secondary circulation during flood tide with near-bed and near-surface currents toward the outer and inner banks of the bend, respectively. The direction of these cross-stream currents was reversed at the end of flood tide. The occurrence and shutdown of these observed unexpected secondary circulation were further investigated through the shape of the along-channel velocity profile by using a theoretical solution based on the steady-state momentum equation and the Stokes's second problem for oscillating flow. Also, an advanced numerical model was used to quantify the unexpected secondary flow. It was shown from the theoretical analysis that vertical advection plays a great role in the momentum balance affecting the direction of secondary flows. The typical secondary flow was found to occur throughout ebb tide with a pronounced near-surface maximum velocity profile. Whereas, during flood tide, the longitudinal velocity profile was characterized by a near-bed and mid-depth maximum, resulting an upwelling flow near the outer bank. These findings were reconfirmed by numerical simulations that were conducted for the entire estuary.

Implementation of a dynamic equation constraint based on the steady state momentum equations within the WRF hybrid ensemble‐3DVar data assimilation system and test with radar T‐TREC wind assimilation for tropical Cyclone Chanthu (2010)

AbstractProper dynamic equation constraints in data assimilation (DA) systems can help improve balance of analyzed atmospheric state. The formulation of ensemble‐variational DA algorithms allows for easy incorporation of such constraints, but their impacts within such DA systems have been little studied. A dynamic constraint based on the steady momentum equations is incorporated into the WRF (Weather Research and Forecasting) hybrid ensemble‐three‐dimensional variational (3DVar) (En3DVar) DA system as a weak constraint. The constraint aims at improving the coupling and balance among wind and thermodynamic state variables, especially when few state variables are directly observed. The scheme is applied to the assimilation of radar T‐TREC (Typhoon‐Tracking Radar Echo by Correlation) winds at a convection‐allowing resolution, for landfalling typhoon, Chanthu (2010), when it was within the range of a coastal radar. Parallel experiments using the 3DVar and En3DVar with and without the dynamic constraint are run to examine the impact of the constraint. The flow‐dependent ensemble covariance used in En3DVar helps to update unobserved pressure and temperature fields in a dynamically more consistent way compared to the static covariance; the added dynamic constraint produces more accurate pressure within the typhoon. The pressure field improved by the dynamic constraint also leads to better temperature and moisture analyses within the variational minimization through flow‐dependent cross covariance. En3DVar analysis with the dynamic constraint produces the best intensity forecast for the typhoon, in terms of the minimum sea level pressure and maximum surface wind speed. Additional sensitivity experiments examine the impact of the weight of the dynamic constraint.

Computational Analysis of Two Phase Flow Distribution in a Horizonatal Pipe

In this paper, a two phase flow distribution in a horizontal pipe is numerically analyzed by solving one dimensional steady state momentum equation for predicting the pressure drop (∆P), quality of steam at outlet of the pipe (X), void fraction (α). The heat absorbed by the pipe (Q) and the mass flow rate (W) of water are varied over a wide range to investigate the above parameters. The locations of the two phase mixture are discussed. Pressure drop along the pipe is inconsistent for different flow rate so, the stable and unstable steady state solution is also carried out using Linear Stability analysis. The present numerical results are compared with the reported data from the literature and found that they are in good agreement. This study is used to calculate the pressure, temperature, hold up and quality within the horizontal pipe.

Bingham plastic fluid flow model in tape casting of ceramics using two doctor blades – analytical approach

One of the most common processes used in manufacturing of multilayer ceramic packages, multilayer capacitors and large scale integration circuits is tape casting. In this process, the wet tape thickness is one of the single most determining parameters affecting the final properties of the product, and it is therefore of great interest to be able to control it. One way to control the tape thickness is to use a two doctor blade configuration in the tape casting machine. In this case, it becomes important to fix the height of the slurry in front of both doctor blades according to the desired tape thickness and casting speed (belt velocity). In the present work, the flow in both doctor blade regions of a slurry is described with a steady state momentum equation in combination with a Bingham plastic constitutive equation, and this is integrated to a closed form analytical solution for both reservoirs based on the desired wet tape thickness and casting speed. The developed model is used to investigate the impact of different material parameters and machine designs on the required slurry height. The solution is compared with experimental findings from the literature, and good agreement is found.

P1/P0+ elements for incompressible flows with discontinuous material properties

We present a 3-noded triangle and a 4-noded tetrahedra with a continuous linear velocity and a discontinuous linear pressure field formed by the sum of an unknown constant pressure field and a prescribed linear field that satisfies the steady state momentum equations for a constant body force. The elements are termed P1/P0+ as the “effective” pressure field is linear, although the unknown pressure field is piecewise constant within each element. The elements have an excellent behavior for incompressible viscous flow problems with discontinuous material properties formulated in either Eulerian or Lagrangian descriptions. The necessary numerical stabilization for dealing with the inf–sup condition imposed by the incompressibility constraint and high convective effects (in Eulerian flows) is introduced via the Finite Calculus (FIC) approach. For the sake of clarity, the element derivation is presented first for the simpler Stokes equations written in the standard Eulerian frame. The extension of the formulation to the Navier–Stokes equations written in the Eulerian and Lagrangian frameworks is straightforward and is presented in the second part of the paper.The efficiency and accuracy of the new P1/P0+ triangle is verified by solving a set of incompressible multifluid flow problems using a Lagrangian approach and a classical Eulerian description. The excellent performance of the new triangular element in terms of mass conservation and general accuracy for analysis of fluids with discontinuous material properties is highlighted.

Self-gravitating equilibrium models of dwarf galaxies and the minimum mass for star formation

We construct a series of model galaxies in rotational equilibrium consisting of gas, stars, and a fixed dark matter (DM) halo and study how these equilibrium systems depend on the mass and form of the DM halo, gas temperature, non-thermal and rotation support against gravity, and also on the redshift of galaxy formation. For every model galaxy we find the minimum gas mass M_g^min required to achieve a state in which star formation (SF) is allowed according to contemporary SF criteria. The obtained M_g^min--M_DM relations are compared against the baryon-to-DM mass relation M_b--M_DM inferred from the \LambdaCDM theory and WMAP4 data. Our aim is to construct realistic initial models of dwarf galaxies (DGs), which take into account the gas self-gravity and can be used as a basis to study the dynamical and chemical evolution of DGs. Rotating equilibria are found by solving numerically the steady-state momentum equation for the gas component in the combined gravitational potential of gas, stars, and DM halo using a forward substitution procedure. We find that for a given M_DM the value of M_g^min depends crucially on the gas temperature T_g, gas spin parameter \alpha, degree of non-thermal support \sigma_eff, and somewhat on the redshift for galaxy formation z_gf. Depending on the actual values of T_g, \alpha, \sigma_eff, and z_gf, model galaxies may have M_g^min that are either greater or smaller than M_b. Galaxies with M_DM \ga 10^9 M_sun are usually characterized by M_g^min \la M_b, implying that SF in such objects is a natural outcome as the required gas mass is consistent with what is available according to the \LambdaCDM theory. On the other hand, models with M_DM \la 10^9 M_sun are often characterized by M_g^min >> M_b, implying that they need much more gas than available to achieve a state in which SF is allowed. Abridged.

Frictional flow of damp granular material in a conical centrifuge

An analysis is given of velocity and pressure-dependent sliding flow of a thin layer of damp granular material in a spinning cone. Integral momentum equations for steady state, axi-symmetric flow are derived using a boundary layer approximation. These reduce to two coupled first-order differential equations for the radial and circumferential sliding velocities. The influence of viscosity and friction coefficients and inlet boundary conditions is explored by presentation of a range of numerical results. In the absence of any interfacial shear traction the flow would, with increasing radial and circumferential slip, follow a trajectory from inlet according to conservation of angular momentum and kinetic energy. Increasing viscosity or friction reduces circumferential slip and, in general, increases the residence time of a particle in the cone. The residence time is practically insensitive to the inlet velocity. However, if the cone angle is very close to the friction angle then the residence time is extremely sensitive to the relative magnitude of these angles.

On three forms of momentum equation in transient modeling of residential refrigeration systems

The transient model of thermal components usually consists of three basic governing equations: continuity equation, energy equation, and momentum equation. For different reasons, the momentum equation is used in three different forms: transient, steady-state, and zero pressure drop. In this paper, the three forms of momentum equation are numerically studied and compared at both component level and system level, particularly in the residential applications. It is found that the results with transient and steady-state momentum equation are pretty close at both levels. Therefore, the steady-state momentum equation is recommended in refrigeration system and component transient modeling because of the robustness and the computation speed. To consider only the system-level cases, it is also found that the transient results without pressure drop give very close estimation of control parameters in comparison with the results taking account into the transient or steady-state pressure drop.

A simple and efficient error analysis for multi-step solution of the Navier–Stokes equations

A simple error analysis is used within the context of segregated finite element solution scheme to solve incompressible fluid flow. An error indicator is defined based on the difference between a numerical solution on an original mesh and an approximated solution on a related mesh. This error indicator is based on satisfying the steady-state momentum equations. The advantages of this error indicator are, simplicity of implementation (post-processing step), ability to show regions of high and/or low error, and as the indicator approaches zero the solution approaches convergence. Two examples are chosen for solution; first, the lid-driven cavity problem, followed by the solution of flow over a backward facing step. The solutions are compared to previously published data for validation purposes. It is shown that this rather simple error estimate, when used as a re-meshing guide, can be very effective in obtaining accurate numerical solutions. Copyright © 2002 John Wiley & Sons, Ltd.

A simple and efficient error analysis for multi‐step solution of the Navier–Stokes equations

AbstractA simple error analysis is used within the context of segregated finite element solution scheme to solve incompressible fluid flow. An error indicator is defined based on the difference between a numerical solution on an original mesh and an approximated solution on a related mesh. This error indicator is based on satisfying the steady‐state momentum equations. The advantages of this error indicator are, simplicity of implementation (post‐processing step), ability to show regions of high and/or low error, and as the indicator approaches zero the solution approaches convergence. Two examples are chosen for solution; first, the lid‐driven cavity problem, followed by the solution of flow over a backward facing step. The solutions are compared to previously published data for validation purposes. It is shown that this rather simple error estimate, when used as a re‐meshing guide, can be very effective in obtaining accurate numerical solutions. Copyright © 2002 John Wiley & Sons, Ltd.

A two-dimensional response to a tropical storm on the Gulf of Mexico shelf

Surface current data from drifting buoys and remotely sensed wind data recorded over the continental shelf in the northeastern Gulf of Mexico during the passage of tropical storm Josephine in October 1996 are examined. Drifter data show the existence of a strong surface jet (velocities reaching 1 m s −1) that moves up the west Florida shelf and westward along the Louisiana–Texas shelf, and lasts for nearly 1 week. The coastal jet occurs during an intense synoptic scale wind event where wind speeds reach 15 m s −1. A simple force balance and statistical analysis are performed to assess the role of strong wind forcing. The primary balance shows an Ekman-type current. The role of local acceleration is greatest when winds are directed along bathymetry. A simple two-dimensional strongly forced shelf response model developed from the linear steady-state momentum equations also indicates larger along-shore currents due to both Ekman-type forcing by cross-shore winds and a cross-shore pressure gradient arising from conservation of mass. Model parameters fit empirically are within 15% of theoretical values. The simple model explains 30% and 46% of the variance in the observed along-shore and cross-shore surface currents, respectively.

Biologically induced circulation at fronts

Consider a frontal region that has high phytoplankton biomass on one side and low biomass on the other. Irradiance penetrates deeply through the water column on the low‐biomass side but is attenuated nearer the surface on the biomass‐rich side because of absorption by phytoplankton. Thus the near‐surface water is heated more on the biomass‐rich side than on the clearer side, resulting in lower‐density surface water on the biomass‐rich side. At greater depths the situation is reversed, with lower‐density water occurring on the biomass‐poor side. We model this situation and examine the resulting perturbations to the frontal circulation. Our aim is to provide an order‐of‐magnitude estimate of the feedbacks from the biological component of the ecosystem to the current field. The model consists of the steady state momentum equations, including Coriolis, pressure gradient, and viscous effects. We compute induced vertical velocities of up to 0.2 mm s−1, commensurate with field measurements and previous modeling estimates of vertical velocities at fronts. The horizontal along‐frontal velocities are of order 2 cm s−1 or less and so will not represent a major contribution to the overall flow field; however, such values are certainly not insignificant.

Can steady-state momentum equations be used in modelling pressurization of adsorption beds?

Modelling of pressurization and blowdown steps of a pressure swing adsorption (PSA) cycle has been carried out using a steady-state momentum equation (Darcy or Ergun) locally in the bed. There has been some controversy about the validity of such an approach since Darcy's or Ergun's equations are strictly valid only for quasi-stable flow, and during pressurization at short times, the flow is unstable. In order to check the assumptions usually made in modelling isothermal pressurization, three models are developed. Model I includes mass balance and unsteady-state mechanical energy balance for a packed bed with non-porous, inert particles; Model II neglects the kinetic energy change in the mechanical energy balance and Model III considers a steady-state mechanical energy balance. The latter model is the one used in previous work. The numerical tool used to solve all the model equations is the moving finite element method (MFEM) with polynomial approximation of any degree in each element developed by C. Sereno 1. Comparison of simulation results for the various models showed that steady-state momentum equations can be safely used in modelling pressurization of adsorbers.

Stress profiles in rail-gun projectiles with distributed armatures

The one-dimensional steady-state momentum equation for rail guns is used to develop an analytic model for the acceleration stresses which arise in rail-gun projectiles. The model is applicable to the case where several armatures are embedded throughout the projectile and provides an expression for the stress as a function of integrals of the electrical conductivity and density distributions in the projectile. This expression is used to derive the conductivity profile which minimizes the stress in the projectile. Stress profiles are calculated for several distributed armature designs. The investigation indicates that, for both solid and arc armatures, the peak stress in a projectile with N uniformly distributed, discrete armatures is proportional to 1/N.

Neoclassical current in a toroidally-confined multispecies plasma

The parallel currents in a toroidally-confined plasma is shown to consist of the current in response to the parallel electric field, the Pfirsch–Schlüter current which is required by charge neutrality, and the bootstrap current which flows to maintain a balance between the viscous and friction forces along the magnetic field. The steady state momentum equation is used to obtain an Ohm’s law relating the neoclassical current to the cross-field fluxes of particles and heat in a multiple ion species plasma. It is found that the presence of impurities does not alter the magnitude of the bootstrap current. Thus, the neoclassical beta limit established by the bootstrap current is insensitive to the impurity content of the plasma. The magnitudes of the bootstrap current and conjugate Ware fluxes are calculated in the transition from the long mean-free-path to collisional regime.