Object Oriented Programming and Numerical Methods

In a tidal marsh plain, salinity and moisture distribution determine the sustainability of the ecosystem. These response patterns are modeled by the marsh plain equations, the governing equations for coupled flow and salinity transport in a saturated/unsaturated tidal marsh plain. The boundary conditions, whose combination is unique to a tidal marsh plain, are established, and evaporation and periodic inundation in a onedimensional soil column is explored. INTRODUCTION In a tidal marsh plain, salinity and moisture distribution determine habitat suitability for local fauna and flora. When the marsh plain is infrequently inundated, large volumes of water evaporate from the surface, creating a dry and hypersaline environment that may be incapable of supporting vegetation. In the creation or restoration of tidal wetlands, the ability to predict when and where such conditions may exist is a crucial factor in success. The unsteady mass conservation equation (Bear 1972), with Darcy’s equations defining the flow velocities, models subsurface flow. The advection-diffusion equation (Fischer et al. 1979), including variable soil saturation, is used to model salinity transport. The governing equations are coupled because fluid density varies with both pressure head and salinity concentration, and the equations must be solved simultaneously. The marsh plain equations are modeled using the numerical technique Method of Lines. A complete discussion of the physics and numerical methods can be found in Greenblatt (1997) and Greenblatt and Sobey (1997). MARSH PLAIN EQUATIONS The marsh plain equations are the governing field equations for coupled flow and salinity transport in the tidal marsh plain, where both saturated and unsaturated flow regions are present (Greenblatt 1997). The full equations are reduced to 1Dept. of Civil & Environmental Engineering, Univ. of California, Berkeley 2Professor, Dept. of Civil & Environmental Engineering, Univ. of California, Berkeley Copyright ASCE 2004 Wetlands 1998 one spatial dimension. Flow will be modeled in the vertical (z) direction, with z positive upwards. The marsh plain flow equation is and the marsh plain salinity transport equation is where p(h, S) [gm/cm3] is the fluid density, K(h) [cm/s] is the hydraulic conductivity, 13(h) is th e moisture content, c(h) [ cm-‘] is the soil moisture capacity, and E [cm2/s], is the diffusion coefficient. The dependent variables are pressure head h = h(z, t) [cm] and salinity S = S(x, t) [psu]. BOUNDARY CONDITIONS The combination of boundary conditions applied to the marsh plain equations is unique to the tidal wetland. Because flow and salinity transport is boundary driven, proper formulation of the boundary conditions is a crucial part of the numerical model. INFILTRATION Infiltration is a downward flux at the surface of the soil column. While the soil surface remains unsaturated, applied water will infiltrate into the soil column. If the applied rate exceeds the infiltration rate, the soil surface becomes saturated, water ponds on the surface, and the actual infiltration rate will be less than the applied rate. A finite control volume approach is implemented to apply the boundary conditions at the top and bottom of the flow domain. Because subsurface flow and salinity transport is slowly varying, sharp gradients may develop at the boundaries if boundary conditions induce large, local changes in the dependent variables. Sharp gradients can cause spurious results in the numerical solution. The control volume approach at the boundaries avoids this purely numerical difficulty without compromising mass conservation. The control volume method treats each end of the soil column as a boundary layer, and uses the boundary conditions at the top and bottom, along with existing conditions in the soil column, to calculate the control volume averaged values of pressure head and salinity. The known applied water rate, qtop, is applied at the top of the control volume. The flow out of the bottom of the control volume, qbot, is a function of the pressure gradient at that location. From Equation 1, mass conservation for the control volume becomes

Recent research has revealed positive effects of unsteady flow on the development of boundary layers in turbine cascades, especially at conditions with a laminar suction side separation bubble at low Reynolds-numbers. Compared to steady flow a reduction of total pressure loss coefficient over a broad range of Reynolds-numbers has been shown. Taking into account the positive effects of wake-induced transition already during the design process, new high lift bladings with nearly the same low losses at unsteady inlet flow conditions could be achieved. This leads to a reduction of weight and cost of the whole turbine module for a constant stage loading. Unsteady flow in turbomachines is caused by the relative motion of rotor and stator rows. For simulating a moving blade row upstream of a linear cascade in the High Speed Cascade Wind Tunnel of the Universität der Bundeswehr München, a wake generator has been designed and built. The wakes are generated with bars, moving with a velocity of up to 40 m/s in the test section upstream of the cascade inlet plane. Unsteady flow causes the transition on the surface of the suction side of a low pressure turbine blade to move upstream whenever an incoming wake is present on the surface, moreover a laminar separation bubble can be diminished or even suppressed. In order to detect the effects of wakes on the boundary layer development a new hot-wire data acquisition system is required. Due to the fact that hot-wires give a good insight into boundary layer development a new hot-wire data acquisition system has been set up. The anemometry system is able of acquiring four channels simultaneously, therefore being capable of logging a triple hot-wire sensor and a bar trigger simultaneously. One further channel is utilised for a once-per revolution trigger. The once-per revolution trigger is used to start the measurement of one data block. Using the well established ensemble averaging technique, 300 ensembles each consisting of 5 wake passing periods have been acquired. Ensemble averaging can be directly performed without any data reduction. The adaptation of this new hot-wire anemometry data acquisition system to the High Speed Cascade Wind Tunnel of the Universität der Bundeswehr München is pointed out. First results on unsteady periodic boundary layer development of a highly loaded low pressure turbine cascade under unsteady inlet flow conditions are presented. During the present investigation four boundary layer traverses, ranging from x/lax = 0.82 to x/lax = 0.99 (suction side), at steady and unsteady inlet flow conditions (Ubar = 10 m/s) at an outlet Reynolds-number of Re2th = 100000 have been conducted.

Related non-stationary formulations of thermoelasticity problems arise in many fields of science and engineering. Analytical solutions are obtained only under significant assumptions, including reduction in the dimensionality of the problem, so numerical methods are required for applied problems, along with those software packages that check reliability procedures of verification andvalidation. Verification is understood as checking the correctness of hypotheses and the formulation of the mathematical formulation, setting correct initial and boundary conditions, choosing a discrete analog and a numerical solution method, taking into account sources of errors and faults. The verification is confirmed by a sufficiently accurate correspondence of the numerical solution to the reference model. The relevance lies in the choice of a suitable reference model. In the present work, the reference model for the thermoelasticity problem is the classical Thompson formula, which describes the temperature change during elastic deformation of a solid body. The error of the numerical solution for the reference problem was of the order of 1% for five characteristic strain values from 0.01 to 0.05. Validation complements the verification procedure and is based on comparison with reliable experimental data or, in their absence, with known analytical solutions. The aim of the work is to carry out verification and validation procedures for the numerical solution of the unsteady problem of thermoelasticity of a deformable solid. The finite element method in the Comsol Multiphysics application package was used. A satisfactory correspondence between the numerical solution and the known analytical solution for the nonlinear heat conduction equation was obtained.

In last two decades, several studies have been conducted to understand the molecular mechanism of slip of fluids (both liquids and gases) at a solid wall. Most of these studies have dealt with steady flow. Recently, Thalakkottor and Mohseni have shown by molecular dynamics simulations that in unsteady flow there is an additional slip above the slip observed in the steady flow. They have developed an unsteady slip flow model by extending the Maxwell’s slip theory for steady flows to encapsulate unsteady flows. The model indicates that the slip velocity of a fluid in unsteady flow is also a function of the acceleration of the fluid in addition to its shear rate. Thus the slip velocity is both a function of firstand second-order derivatives normal to the wall multiplied by different coefficients dependent upon the nature of the fluid (liquid or gas). In this paper, we consider several unsteady flows – the Stokes flow, the Couette flow, and flow in a channel due to an oscillatory wall. In addition, we consider unsteady flow due to an impulsively started flat plate. Exact solutions are obtained by using the most general form of slip boundary condition that contains both first and second-order derivatives normal to the boundary. The coefficients that multiply these derivatives can be determined by knowledge of the nature of the fluid. It should be noted that these solutions can also be reduced to steady flows with slip with higher-order slip boundary condition that includes both firstand second-order derivatives normal to the wall.

The present work forms part of a research project of the Institute of Jet Propulsion and Turbomachinery at the RWTH Aachen University in collaboration with GE Aviation. The subject is the detailed numerical analysis of the unsteady flow field, focusing on the interaction between the impeller and the passage diffuser of a close-coupled transonic centrifugal compressor used in an aero engine. The centrifugal compressor investigated is characterized by a close-coupled impeller and passage diffuser with a radial gap of only 3.6%. The close coupling tends to provide a high aerodynamic efficiency but simultaneously cause a high unsteady interaction between the impeller and the diffuser. These unsteady effects can have a significant impact on the performance of both components and present a challenge to state-of-the-art numerical methods. With increasing compressor efficiency, the more important it is to have an understanding of the detailed unsteady flow physics. Experimental data was obtained from a state-of-the art centrifugal compressor test rig located at the Institute of Jet Propulsion. Steady and unsteady pressure measurements within the impeller and diffuser are used to gain detailed information on the temporal, time-averaged, and spectral pressure distributions within the stage to validate the CFD. The work presented here shows the unsteady phenomena caused by the interaction and the location and propagation of these phenomena within the centrifugal stage. Within the impeller, the exducer is in first order excited by the blade passing frequency (BPF) of the diffuser, whereas in the diffuser both the BPF and the passage passing frequency (PPF), are present up until the end of the pipe-diffuser. Significant effects on the integral component performance could only be identified for the impeller. Special focus is paid to evaluate the diffuser upstream pressure field, since this is the major source of unsteadiness within the impeller. The performance of the rotor decreases due to the unsteady interaction. This effect is traced back to the unsteady tip-clearance flow, in which the time-averaged mass transport decreases, whereas the specific entropy production increases in a nonlinear way. Within the diffuser, local effects counteracting with respect to the integral performance are found. In front of the throat, there is less decay in the total pressure as a result of tangentially expanding pressure waves. Within the passage a decrease in flow uniformity in the unsteady flow is identified as the reason for the lower diffusion up until the throat and higher losses within the downstream diffuser passage.

The present work forms part of a research project of the Institute of Jet Propulsion and Turbomachinery at the RWTH Aachen University in collaboration with GE Aviation. The subject is the detailed numerical analysis of the unsteady flow field focusing on the interaction between the impeller and the passage diffuser of a close-coupled transsonic centrifugal compressor used in an aero engine. The centrifugal compressor investigated is characterized by a close-coupled impeller and passage diffuser with a radial gap of only 3.6%. The close coupling tends to provide a high aerodynamic efficiency but simultaneously cause a high unsteady interaction between the impeller and the diffuser. These unsteady effects can have a significant impact on the performance of both components [1,2] and present a challenge to state-of-the-art numerical methods. With increasing compressor efficiency, the more important it is to have a understanding of the detailed unsteady flow physics. Experimental data was obtained from a state-of-the art centrifugal compressor test rig located at the Institute of Jet Propulsion [3]. Steady and unsteady pressure measurements within the impeller and diffuser are used to gain detailed information on the temporal, time-averaged and spectral pressure distributions within the stage to validate the CFD. The work presented shows the unsteady phenomena caused by the interaction as well as the location and propagation of these phenomena within the centrifugal stage. Within the impeller, the exducer is in first order excited by the BPF of the diffuser, whereas in the diffuser both the BPF, as well as the PPF, are present up until the end of the pipe-diffuser. Significant effects on the integral component performance could only be identified for the impeller. Special focus is paid to evaluate the diffuser upstream pressure field, since this is the major source of unsteadiness within the impeller. The performance of the rotor decreases due to the unsteady interaction. This effect is traced back to the unsteady tip-clearance flow, in which the time-averaged mass transport decreases, whereas the specific entropy production increases in a nonlinear way. Within the diffuser, local effects counteracting with respect to the integral performance are found. In front of the throat, there is less decay in total pressure, as a result of tangentially expanding pressure waves. Within the passage an decrease in flow uniformity in the unsteady flow is identified as the reason for the lower diffusion up until the throat and higher losses within the downstream diffuser passage.

Study and application of steady flow and unsteady flow mathematical model for channel networks

We consider an enzyme-substrate reaction-diffusion problem. Unsteady and steady state models are recalled. For the unsteady state case, the model is in the form of a second order partial differential equation. We solve the unsteady state model using the explicit numerical finite difference method, which is forward difference in time and centered difference in space. For the general steady state case, the model is in the form of a second order ordinary differential equation. We solve the general steady state model using the explicit first order Euler's numerical method. For the particular steady state case of the unsaturated catalytic kinetics, we derive the exact analytical solution using the characteristic method of ordinary differential equations. For the particular steady state case of the saturated catalytic kinetics, we derive the exact analytical solution using the direct-integration method. The obtained exact analytical solutions are identical with the existing exact analytical solutions derived using the variational iteration method. With the aid of computer, the enzyme-substrate reaction-diffusion problem can be solved and simulated successfully for both unsteady and steady state cases.

Since 1995, the Illinois State Water Survey (ISWS) has conducted a series of studies on unsteady flow modeling of the lower Illinois River using the one-dimensional unsteady state flow (UNET) model with a primary focus on the management options of Levee and Drainage Districts (LDDs) for flood peak reduction should overtopping or levee failure occur. The same model also is being used as a tool for evaluating impacts of proposed or ongoing restoration efforts in the Illinois River basin. The Illinois River is one of the major tributaries of the Mississippi River. Most of its floodplains from Peoria to Grafton have been leveed for agricultural purposes through the establishment of LDDs. A total of 24 LDDs were established between 1879 and 1916. The UNET model uses either stage or flow hydrographs as boundary conditions. The Hydraulic Engineering Center’s storage database (HEC-DSS) is used in the UNET model for input and output hydrographs and hydrograph ordinates, which enables easily adapting the model for large river systems and various scenarios. For the lower Illinois River, the UNET model was used to evaluate the impact of using a single LDD or combinations of selected LDDs to reduce peak stages. Spillways of certain sizes were assumed to be placed for floods through the levees. The UNET model also is being used to evaluate the possible reduction of flood stage fluctuations during the growing season by opening selected LDDs or by removing the levees to open a portion or all floodplain areas to the river. It is generally assumed that a well-calibrated and verified hydraulic model will be sufficient for evaluating management and restoration alternatives that involve possible modifications of model geometries. Application of the UNET model for the lower Illinois River showed that the stage and flow relations can be changed drastically by modifying the channel geometries, however. Boundary conditions for the UNET model have to be adjusted accordingly to maintain mass balance for the system. This study presents comparisons of stage and flow relations altered by possible restoration efforts under assumed scenarios, and the problems caused when appropriate boundary conditions were not used in the UNET model for the Illinois River system. Copyright ASCE 2004 World Water Congress 2004

Non-equilibrium thermodynamics is a relatively new field of research, and one which is also becoming increasingly important in engineering applications. However, despite the many approaches developed in recent decades, they all raise questions from a mathematical point of view, for instance on the proper definitions of initial and boundary conditions, analytical and numerical solution methods, and geometrical background. These questions cover numerous topics, including the mathematical analysis of various thermodynamic approaches, the investigation of the resulting governing equations, symmetry analysis of the solutions, and how the numerical methods can be built on the geometrical background. The aim of this Special Issue is to discuss and present up-to-date problems that may not be constrained to the previously mentioned aspects but must be connected to non-equilibrium thermodynamics. Thus, in the present Special Issue we aim to present the open mathematical questions on and characteristics of different thermodynamic approaches.

This paper presents results of unsteady viscous flow calculations and corresponding cold flow experiments of a three-stage low-pressure turbine. The investigation emphasizes the study of unsteady flow interaction. A time-accurate Reynolds-averaged Navier–Stokes solver is applied for the computations. Turbulence is modeled using the Spalart–Allmaras one-equation turbulence model and the influence of modern transition models on the unsteady flow predictions is investigated. The integration of the governing equations in time is performed with a four-stage Runge–Kutta scheme, which is accelerated by a two-grid method in the viscous boundary layer around the blades. At the inlet and outlet, nonreflecting boundary conditions are used. The quasi-three-dimensional calculations are conducted on a stream surface around midspan, allowing a varying stream tube thickness. In order to study the unsteady flow interaction, a three-stage low-pressure turbine rig of a modern commercial jet engine is built up. In addition to the design point, the Reynolds number, the wheel speed, and the pressure ratio are also varied in the tests. The numerical method is able to capture important unsteady effects found in the experiments, i.e., unsteady transition as well as the blade row interaction. In particular, the flow field with respect to time-averaged and unsteady quantities such as surface pressure, entropy, and skin friction is compared with the experiments conducted in the cold air flow test rig. [S0889-504X(00)02004-3]

The breakdown for both an isolated vortex and a vortex flow in a tube is studied by means of a numerical solution of the axisymmetric Navier-Stokes equations for an unsteady, incompressible, and laminar flow. Results are compared to those of other authors and to experiments. Various numerical solution methods and various physical boundary conditions are used in order to explain the existence of different solutions. The computational results reveal that the existence of different solutions is due to slightly different inflow conditions. The numerical problem for the isolated vortex becomes ill-posed at a certain time for the inflow condition of zero axial gradient of the radial velocity component. Different inflow conditions yield breakdown flowfields containing different features of the experimentally observed breakdown structure.

Very few studies have been carried out in the past in estimating depth-averaged velocity and bed shear stress in unsteady flow over rough beds. An experiment is thus conducted to investigate the vertical and lateral velocity profiles under unsteady flow conditions in a rough open channel for various flow depths. One hydrogram is repeatedly passed through the rectangular flume with a fixed rigid grass bed. Using micro Pitot tube and Acoustic Doppler Velocimeter (ADV), the flow patterns are investigated at both lateral and longitudinal positions over different cross-sections. For two typical flow depths, the velocities in both the rising limb and falling limb are observed. Hysteresis effect between stage-discharge (h ~ Q) rating curve between rising and falling limbs is illustrated. Lateral distribution of depth-averaged velocity and bed shear stress are plotted at three different cross sections and compared with the steady flow conditions. In falling limb of an unsteady flow case, both depth-averaged velocity and bed shear stress distribution in the central region is higher than that of steady flow case. However, in the rising limb, the bed shear stress of unsteady flow is less than that of steady flow case. Further, in an unsteady flow, the magnitude of depth-averaged velocity is found to increase towards the downstream sections. Along the downstream positions, bed shear stress values increase for lower flow depths and decrease for higher flow depth cases.

A Navier-Stokes code with a low Reynolds number k-.epsilon. turbulence model was tested to investigate its predictability for the unsteady transitional boundary layer flow due to rotor-stator interaction. A preliminary calculation with three different numbers of time steps 300, 600, and 1000 for a rotor wake passing period was carried out to see the effects of time steps on the unsteady flow and pressure fields due to rotor-stator interaction. Numerical solutions showed that unsteady pressure was much more sensitive to the number of time steps and over 600 time steps should be used to get a numerical solution independent of the number of time steps for a rotor wake passing period. The original low Reynolds number k-.epsilon. turbulence model showed very poor prediction of the unsteady transitional boundary layer flow due to rotor-stator interaction. This was due to the excessive production of turbulent kinetic energy near the leading edge. A modification suggested by Launder was incorporated and the modified model captured well the wake induced transitional strip. Present solutions also showed improved prediction over previous Euler/boundary layer solution in terms of the onset of unsteady transition and its extent.

A numerical method is presented for calculating the pressure, forces and moments on arbitrary-shaped airfoils in an unsteady potential flow between two parallel walls. Airfoils and walls are represented by vorticity distributions and mirror images respectively. Vortex movements are determined by discrete vortex method. Our method does not have the restrictions on the height of walls and the location of airfoils between walls. The calculations are carried out for two cases of starting flow and accelerating flow. It is found that when walls come closer, the steady lifts, moments and apparent mass become greater, and the unsteady lifts and moments reach the steady values faster, and that the pressure center of airfoils moves notably under the influence of the flow acceleration.