Search

The Rayleigh-Benard convection of viscoelastic fluids in a cavity is simulated using the grid-by-grid inversion method. The grid-by-grid inversion method allows the algorithms for Newtonian fluids to be applicable to viscoelastic fluids using a technique of deferred correction. Until now, the detailed characteristics of thermal convection in a cavity have not been investigated sufficiently. Employing the grid-by-grid inversion method, we have found a peculiar phenomenon specific to the viscoelastic fluids, which cannot be predicted by the linear and nonlinear stability analysis. Contrary to the cases of Newtonian fluids and viscoelastic fluids in the parameter range of exchange of stabilities, the convection cell of viscoelastic fluids in the parameter range of Hopf bifurcation changes the circulation direction at each temporal period of oscillatory convection.

We investigate forced convection of a nanofluid in the entrance region of a cylinder and in the fully-developed region of a coaxial cylinder employing the pseudo-single-phase continuum model and the homogeneous single-phase model. While the homogeneous single-phase model assumes a uniform nanoparticle distribution, the pseudo-single-phase continuum model takes care of nonuniform nanoparticle distribution. There has been controversy regarding the cause of heat transfer enhancement in nanofluids. Whether it is caused solely by the variation of thermophysical properties or the nanoparticle distribution also affect it. This controversy may be resolved employing these two models. It is found that nanoparticles drift from hot wall to cold central region in the entrance region, while they drift from cold inner surface to hot outer surface in the fully-developed coaxial cylinder due to the thermophoresis. The resulting nonuniform distributions of nanoparticles are found to add the heat transfer enhancement slightly. On the other hand, the frictional dissipation increases when the heat transfer rate increases. In a sense, the enhanced heat transfer rate is partly achieved at the expense of a higher energy consumption.

The packed bed may be viewed as consisting of two distinct fields, i.e., one existing on the macroscale and the other existing on the microscale levels. The macroscale describes phenomena occurring in the bed dimension and the microscale for those occurring in the intraparticle dimension. The macroscale temperature and concentration fields are found from a hypothetical continuum model of particle-fluid heterogeneous media. Each macroscale differential volume element contains several catalyst pellets, each of which is a microscale continuum embedded in a gross macroscale external fields. The boundary conditions of the microscale fields at the catalyst surfaces are provided by the solution of the macroscale model equations, while the heat and mass source terms of the macroscale model equations are determined by the microscale heat and mass transfer and chemical reactions within the catalyst pellets. In the present investigation, we have considered the HPPO process in which the propene oxide is produced by oxidizing propene with hydrogen oxide using the titanium silicalite-1(TS-1) catalysis. The set of six partial differential equations for the macroscale field coupled with the set of five partial differential equations for the microscale field at each macroscale point is solved employing the numerical Green’s function technique to overcame the inherent numerical instability in this coupled PDE system. The effects of pellet size, intra-pellet diffusivity and conductivity on the reactor yield and catalyst effectiveness factor are investigated.

Nanofluid is a suspension of nanoparticles with high thermal conductivity. Many investigations have been performed to find out the mechanism of enhanced heat transfer in the nanofluids. In the present work, we model the Al 2 O 3 /H 2 O nanofluid employing the pseudo-single-phase continuum model and investigate the fully-developed forced convection in a coaxial cylinder under a fixed pressure drop. It is found that the heat transfer coefficients of both the cold inner wall and hot outer wall increase with respect to the inlet mass fraction of nanoparticles. The enhancement of convective heat transfer coefficient is found to be slightly higher than that of thermal conductivity of nanofluid itself due to the drift of nanoparticles. On the other hand, the Nusselt number of cold inner wall increases but that of hot outer wall decreases as the inlet mass fraction of nanoparticles increases. It is revealed that this peculiar behavior of Nusselt number on the hot outer wall is caused by the drift of nanoparticles from the cold inner wall to the hot outer wall due to the thermophoresis. As the inlet mass fraction of nanoparticles increases under the condition of fixed pressure drop, the mixture viscosity increases and the volumetric flow rate is retarded, since the viscous dissipation must balance the input rate of mechanical energy at the steady state.

Recently, Due to the increased industry using radiation inspection equipment in the semiconductor, this demand of technology research is increasing. Although semiconductor inspection equipment is using low energy X-ray from 40 keV to 120 keV, Studies of radiation damage about the low energy X-ray are lacking circumstance in our country. Therefore, It is study that BJT (bipolar junction transistor) of one type of semiconductor elements are received radiation damage by low energy X-ray. BJT were used to the NXP semiconductor company's BC817-25 (NPN type), and Used the X-ray generator for the irradiation. Radiation damage of BJT was evaluated that confirm to analyse change of collector-emitter voltage of before and after X-ray irradiation when current gain fixed to 10. X-ray generator of tube voltage was setting 40 kVp, 60 kVp, 80 kVp, 100 kVp, 120 kVp and irradiation time was setting 180s, 360s, 540s into 180s intervals. As the result, We confirmed radiation damage in BJT by low energy X-ray under 120 keV energy, and Especially the biggest radiation damage was appeared at the 80 kVp. It is expected that ELDRS (enhanced low dose rate sensitivity) phenomenon occurs on the basis of 80 kVp. This studies expect to contribute effective dose administration of semiconductor inspection equipment using low energy X-ray, Also Research and Development of X-ray filter.

Nanofluids are composed of fluids and dispersed submicron solid particles. The presence of nanoparticles in the fluids enhances the effective thermal conductivity. In order to predict the enhancement of heat transfer in the nanofluids, it is necessary to model the nanofluid systems rigorously from the viewpoint of fluid dynamics. In the present investigation, we suggest a fluid mechanical model of nanofluids based on a rigorous theory of continuum mechanics. Starting from a two-fluid model, a pseudo-single-phase model is derived exploiting the fact that the velocity and temperature of nanoparticles follow tightly those of base fluid. The resulting pseudo-single-phase model of nanofluids is employed to investigate the Rayleigh–Benard convection of nanofluids. It is revealed that the presence of nanoparticles retards onset of convection and reduces convective fluid motion. Although the Nusselt number Nu and the heat transfer coefficient h are increasing functions of the Rayleigh number Ra for all values of particle mass fraction ωp, Nu is almost independent of ωp while h increases with respect to ωp because the thermal conductivity increases as ωp increases. The present pseudo-single-phase model of nanofluids may be adopted to predict heat and mass transfer as well as fluid dynamic characteristics in various nanofluid systems.

A multiscale packed bed model is proposed to simulate industrial carbon dioxide absorber and stripper employing the hot potassium carbonate process promoted by piperazine. At a macro scale, a material and energy balance for the packed bed is built with the consideration of basic reaction chemistry, phase equilibria and solution thermodynamics, where the facilitated mass transfer of CO2 caused by chemical reactions is taken care of by an enhancement factor. At a micro scale, the enhancement factor is calculated from the concentration profiles of chemical species in the liquid film around packing materials. The computational burden for the numerical solution of a set of diffusion–reaction equations at each macro scale location is circumvented by means of the Karhunen–Loeve Galerkin (KLG) method. The present multiscale model, where the KLG method treats the micro scale process of diffusion and reaction in the liquid film, is found to yield accurate predictions at a reasonably small computational cost. This method of solving multiscale problems can be applied not only to the present CO2 absorption/desorption processes but also to various systems in science and engineering, where the necessity of multiscale resolution is important.

Carbon dioxide emitted from the power plants may be removed in an absorber-stripper unit, where CO2 mass transfer enhanced by chemical reactions occurs. The mass transfer enhancement factor can be found accurately once we find the CO2 profile in the liquid film around a packing material. However, it costs a tremendous amount of computer time to solve governing diffusion–reaction equations for the relevant chemical species at each grid point of the packed bed to determine the enhancement factor accurately. Therefore, in the traditional models of packed bed, one adopts an approximate enhancement factor found from a simple model. In the present work, we employ the Karhunen–Loeve Galerkin (KLG) method for the solution of the diffusion–reaction equations describing the CO2 absorption with potassium carbonate promoted by piperazine in the liquid film. With the KLG method, the computational time for the diffusion–reaction equations is reduced drastically and the rigorous model for the enhancement factor can be adopted in the industrial packed beds. The resulting reduced order model is found to predict the concentration profiles of chemical species accurately. It shall be adopted in the multiscale model for a packed bed, where the macro scale describes the overall geometry of the packed bed and the micro scale describes the transport and reaction in the thin liquid film around a packing at an arbitrary macro location in the bed.

A practical method is developed to solve inverse natural convection problems sequentially. The sequential algorithm is based on the Kalman filtering technique. By reducing the Boussinesq equation to a reduced order model through the Karhunen–Loeve Galerkin procedure, the computational difficulties associated with the covariance equation are overcome. Although this method is found to yield accurate results with efficiency, its accuracy depends on that of the reduced order model employed. In the present investigation, it is demonstrated that this method can be further developed to be reasonably accurate and computationally feasible even with a relatively inaccurate reduced order model by employing the Boussinesq equation as the state equation while adopting the Kalman filter constructed from less accurate reduced order model.

Disturbances in blood flow to intervertebral discs (IVD) play an important role in IVD degeneration. Vascular endothelial growth factor (VEGF) and endothelial nitric oxide synthase (eNOS) are extremely important angiogenic factors for vasodilation and neovascularization. We investigated the relationship between single nucleotide polymorphisms (SNPs) of the VEGF and eNOS genes and genetic susceptibility to lumbar IVD degeneration in a young adult Korean population. Two hundred and forty-one participants (aged 18 to 30 years), with or without low back pain, were selected for the study. Magnetic resonance imaging was made of the lumbar spine in all participants. The patient group (N = 102) had low back pain clinically and lumbar IVD degeneration radiographically. The control group (N = 139) included subjects with and without low back pain; all were negative radiographically for lumbar IVD degeneration. Using PCR-RFLP analysis, we analyzed VEGF (-2578C>A, -1154G>A, -634G>C, and 936C>T) and eNOS (-786T>C, 4a4b and 894G>T) SNPs. We made combined analyses of the genes and performed haplotype analyses. There were no significant differences in the genotype distribution of polymorphisms of VEGF and eNOS genes among patients and controls. However, the frequency of VEGF -2578CA +AA/-634CC combined genotypes was significantly higher in patients when compared with controls [odds ratio (OR) = 21.00; 95% confidence interval (CI) = 2.590- 170.240]. The frequencies of the -2578A/-1154A/-634C/936C (OR = 3.831; 95%CI = 1.068-13.742), -2578A/-1154A/-634C (OR = 3.356; 95%CI = 1.198-9.400), and -2578A/-634C/936C (OR = 10.820; 95%CI = 2.811-41.656) haplotypes were also significantly higher in patients than in controls. We conclude that the combined genotype VEGF -2578CA+AA/-634CC is a possible risk factor for IVD degeneration and the VEGF -2578A/-1154A/-634C/936C haplotype may increase the risk for development of IVD degeneration. Furthermore, the VEGF -634C allele appears to be associated with susceptibility to IVD degeneration.

Microchannels made of polymeric materials usually have wall velocity slip. Sometimes, these microchannels are not smooth. Wall roughness is introduced by manufacturing processes or caused by adhesion of biological particles from the liquids. For a proper design and operation of these microchannels, it is important to secure an accurate value of zeta potential, which determines the volumetric flow rate of electroosmotic flows. Zeta potential of microchannels is usually determined by measuring the streaming potential. In the present investigation, we have derived a simple formula that predicts accurately the streaming potential of microchannels having wall roughness and Navier velocity slip, and investigated how the streaming potential is affected by wall roughness, slip coefficient, zeta potential and bulk ionic concentration. The simple formula, which requires numerical solution of three sets of Stokes equation sequentially using a small grid number, yields very accurate results as compared with the numerical solution of nonlinear partial differential equations employing a very large number of grids. It is found that the streaming potential per pressure drop increases with respect to slip coefficient and decreases with respect to wall roughness hight. On the other hand, the streaming potential per volumetric flow rate increases as the wall roughness increases.

Cavitation causes a great deal of noise, damage to components, vibrations, and a loss of efficiency in devices, such as propellers, pump impellers, nozzles, injectors, torpedoes, etc., Thus, cavitating flow simulation is of practical importance for many engineering systems. In this study, a two-phase flow solver based on the homogeneous mixture model has been developed. The flow characteristics around an axisymmetric cylinder were calculated and then validated by comparing with the experimental results in the cavitation water tunnel at the Korea Ocean Research & Development Institute. The results show that this solver is highly suitable for simulating the cavitating flows. After the code validation, the cavity length with changes of water depth, angle of attack and velocity were obtained.. Cavitation inception was also calculated for various operational conditions.

In the nonisothermal gaseous flows through open vessels with chemical reaction and combustion processes, the flow can be approximated with compressible Navier–Stokes equation where density depends on temperature but is independent of velocity and pressure fields, which is called the low mach number approximation. To implement advanced estimation and control schemes such as Kalman filtering and optimal feedback control to low Mach number flows, it is necessary to find an accurate and reliable reduced-order model. One of the best methods available for this purpose is the Karhunen–Loeve Galerkin (KLG) procedure. The application of the KLG procedure to low Mach number flows, however, is never straightforward because some variables depend on temperature nonlinearly or exponentially and the velocity field is not solenoidal which prohibits the elimination of pressure in the Galerkin procedure. In the present investigation, we have overcome these difficulties by introducing appropriate auxiliary variables and have successfully derived a reduced-order model for low Mach number flows with radiative heat transfer and chemical reactions. The resulting reduced-order model is found to yield accurate results efficiently. The present work sets up the first stage for the real-time estimation and control of various gaseous nonisothermal reactors and combustors.

Rayleigh–Benard convection of viscoelastic fluids in a cavity is investigated using a newly developed grid-by-grid inversion method. In the grid-by-grid inversion method the hyperbolic constitutive equation is split such that the term for the convective transport of stress tensor is treated as a source. This renders the stress tensor a local function of the velocity gradient tensor as in the case of the Newtonian fluids and makes the algorithms for Newtonian fluids applicable to viscoelastic fluids. To corroborate the accuracy of the grid-by-grid inversion method, a linear stability analysis is performed to find the critical Rayleigh number and the domains of Hopf bifurcation and exchange of stabilities in the parameter space. The numerical results from the grid-by-grid inversion method are found to coincide with those of linear stability analysis exactly. Also considered is the standard benchmark problem of viscoelastic flow past a cylinder placed at the center between two plates to confirm the accuracy of the grid-by-grid inversion method.

We consider electroosmotic flows (EOF) generated by temporally varying zeta potential which is usually adopted for pumping or mixing of fluids. In this case the dynamics of ions in the electric double layer (EDL) influences the induced electric field and consequently the EOF significantly. Therefore, the appropriate model should be the Nernst–Planck (NP) model for all zeta potentials or the Debye–Huckel (DH) model for low zeta potentials rather than the Poisson–Boltzmann (PB) model which is based on the equilibrium distribution of ions in the EDL. In the present investigation, we compare the predictions from the DH model and the PB model with the exact ones from the NP model for a range of frequency of zeta potential oscillation. It is found that one may adopt the PB model when the frequency is low and the DH model when the zeta potential is low. However, for either high frequency of zeta potential oscillation or large value of zeta potential, one must adopt the NP model to get accurate predictions of EOF.

This study was conducted to develop an agaricus bisporus washing machine which uses compressed air to remove foreign materials attached on the surface of agaricus bisporus. A prototype of the washing machine was constructed, and performance of removing foreign materials was tested. Research results are as follows: 1. Several transferring methods including PE roller rotation, brush roller rotation, PE screw rotation, vibration plate, and belt conveyor were evaluated. Roller, screw, and vibration methods caused damages on the surface of the products, but belt conveyor method caused the least damages. 2. For air jet, a stationary nozzle type and a rotational type were evaluated. The best air jet nozzle was the jet-type nozzle, and the rotational type was more effective than stationary type nozzle. 3. With the conveyer belt, box type moving method and the rotational air jet nozzle, the washing machine showed the best performance when higher than 5.4105 Pa of air jet pressure and lower than 0.047 m/s of moving speed was used. Working performance of the system was 650 kg/h, and the damaging rate was 1.2 %.

We have performed the flat-fielding correction for the full-disk monitoring system of KASI (Korea Astronomy and Space Science Institute), which is installed in the Solar Flare Telescope (SOFT) at the top of Bohyun Mountain. For this, we used a new method developed by Chae (2004), to determine the flat pattern from a set of relatively shifted images. Using this method, we successfully obtained the flat pattern for full-disk observations and compared our result with the image observed in Catania Astrophysical Observatory. The method that we used in this study seems to be quite powerful to obtain the flat image for solar observations. In near future, we will apply this method for the flat-fielding correction of all solar imaging instruments in KASI.

A reduced-order model is derived for electroosmotic flows in a microchannel of nonuniform cross-section. The model is constructed based on the Karhunen–Loeve Galerkin (KLG) procedure which can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations. The reduced-order model of the present investigation is carefully constructed such that it is not necessary to re-evaluate any coefficient of the model even though the inhomogeneous zeta potential ζ ( x ) and the dielectric constant e vary. This feature reduces the computational time greatly when employed in the estimation and control of electroosmotic flows, where repeated solution of governing equations is required. Using the present reduced-order model, a practical method is devised to estimate inhomogeneous zeta potential ζ ( x ) from velocity measurements of the electroosmotic flow in the microchannel. The proposed method is found to estimate ζ ( x ) with reasonable accuracy even using noisy velocity measurements.

In the present investigation, we develop a method for estimating rheological parameters of viscoelastic fluids using velocity measurement in a square straight channel. It is believed that a somewhat complicated patterns of secondary flows due to the non-zero second normal stress difference are more useful than the simple viscometric flows traditionally adopted in the determination of rheological parameters. The inverse problem of determining the rheological parameters from a set of velocity measurements is solved using a conjugate gradient method. When applied to a general constitutive equation encompassing the UCM model, the Oldroyd-B model and the PTT model, the present method is found to yield a reasonably accurate estimation of five rheological parameters simultaneously even with noisy velocity measurements.

In the present work, we consider the linear and nonlinear hydrodynamic stability problems of two-dimensional Rayleigh–Benard convection in arbitrary finite domains. The effects of the domain shapes on the critical Rayleigh number and convection pattern are investigated by means of a linear stability analysis employing a Chebyshev pseudospectral method. An extension of the present technique to nonlinear stability analysis allows derivation of the Landau equation for arbitrary finite domains. The results of nonlinear stability analysis are confirmed by comparison with numerical solution of the Boussinesq set. The results of the present investigation may be exploited to enhance or suppress thermal convection by varying system domain.

In the present work, we consider the linear hydrodynamic stability problems of viscoelastic fluids in arbitrary finite domains. The effects of domain shapes on the critical Rayleigh number and convection pattern are investigated by means of a linear stability analysis employing a Chebyshev pseudospectral method. It is shown that the domain shape can change the viscoelastic parameter values where the Hopf bifurcation occurs in the Rayleigh–Benard convection. The results of the present investigation may be exploited to design shapes of convection box where the Hopf bifurcation occurs at realistic low values of Deborah number. This will enhance the usefulness of the natural convection system as a rheometry tool.

A three-dimensional inverse radiation problem is solved sequentially using an extended Kalman filter, where the time-varying strength of a heat source is determined from temperature measurements in the participating media. By employing the Karhunen–Loeve Galerkin procedure that reduces the set of governing equations composed of the radiative transfer equation and heat conduction equation to a small set of ordinary differential equations, the computational difficulties associated with the Kalman filtering for the integro-differential equation are overcome. The present technique allows a real-time implementation of the sequential identification of radiation field in participating media.

The Karhunen–Loeve Galerkin procedure is a type of Galerkin method that employs as basis functions the empirical eigenfunctions of the Karhunen–Loeve decomposition, and proved to be a powerful tool for reducing the degree of freedom of partial differential equations [Int. J. Numer. Methods Engrg. 41 (1998) 1131; Comput. Methods Appl. Mech. Engrg. 188 (2000) 165]. In the present investigation, we extend the applicability of the Karhunen–Loeve Galerkin procedure to systems of variable domains and apply it to the shape identification problems of heat conduction systems, where the unknown boundary shape is estimated from temperature measurements on the other boundary. The efficiency and accuracy of the present technique are assessed as compared to the traditional technique employing the original partial differential equation.

In the radiant cooler, where the hot gas from the pulverized coal gasifier or combustor is cooled to generate steam, the wall heat-transfer coefficient varies due to ash deposition. In the present work, we investigate an inverse radiation problem of estimating the heat-transfer coefficient from temperature measurement in the radiant cooler. The inverse radiation problem is solved through the minimization of a performance function, which is expressed by the sum of square residuals between calculated and observed temperature, utilizing the conjugate gradient method. The gradient of the performance function is evaluated by means of the improved adjoint variable method, which resolves the difficulty associated with the singularity of the adjoint equation through its inherent regularization property. The effects of the number of measurement points and measurement noise on the accuracy of estimation are also investigated.

The control of natural convection in magneto-hydrodynamic (MHD) flow is investigated by means of the Karhunen–Loeve Galerkin procedure [Int J Numer Meth Engng 1998;41:1133–51]. The Karhunen–Loeve Galerkin procedure, which is a type of Galerkin methods that employs the empirical eigenfunctions of the Karhunen–Loeve decomposition as basis functions, can reduce non-linear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. In the present investigation, it is demonstrated that the Karhunen–Loeve Galerkin procedure is well suited for the problems of control or optimization, where one has to solve the governing equations repeatedly but one can also estimate the approximate solution space from the range of control variable. The performance of the Karhunen–Loeve Galerkin procedure for solving the optimal control problem of natural convection is assessed in comparison with the traditional technique employing the Boussinesq equation, and is found to be very accurate as well as efficient.

Protein–nucleic acid interaction is an important process in many biological phenomena. In this study, a fluorescence resonance energy transfer (FRET)-based protein–DNA binding assay has been developed, in which a fluorescent amino acid is genetically incorporated into a DNA-binding protein. A coumarin-containing amino acid was incorporated into a DNA-binding protein, and the mutant protein specifically produced a FRET signal upon binding to its cognate DNA labeled with a fluorophore. The protein–DNA binding affinity was then measured under equilibrium conditions. This method is advantageous for studying protein-nucleic acid interactions, because it is performed under equilibrium conditions, technically easy, and applicable to any nucleic acid-binding protein.

An applicable method is developed for the identification and feedback control of natural convection. The Boussinesq equation is reduced to a small set of ordinary differential equations by means of the Karhunen–Loeve Galerkin procedure [Int. J. Heat Mass Transfer 39 (1996) 3311]. Based on this low-dimensional dynamic model, a feedback control synthesis is constructed by first performing an extended Kalman filter estimate of the velocity and temperature fields to treat the measurement errors and then developing the optimal feedback law by means of the linear quadratic regulator theory. The present method allows for the practical implementation of modern control concepts to many flow systems including natural convection.

We consider an inverse radiation problem of determining the time-varying strength of a heat source, which mimics flames in a furnace, from temperature measurements in three-dimensional participating media where radiation and conduction occur simultaneously. The inverse radiation problem is posed as a minimization problem of the least-squares criterion, which is solved by a conjugate gradient method employing the adjoint equation to determine the descent direction. The discrete ordinate S4 method (M.F. Modest, Radiative Heat Transfer, McGraw-Hill, New York, 1993) is employed to solve the radiative transfer equation and its adjoint equation accurately. The performance of the present technique of inverse analysis is evaluated by several numerical experiments, and it is found to solve the inverse radiation problem accurately without a priori information about the unknown function to be estimated.

Rapid thermal processing (RTP) is a new technique for performing various wafer fabrication operations such as annealing, oxidation and chemical vapor deposition in a single chamber. The success of the RTP depends on the precise control of wafer temperature by adjusting wall heat flux. In the present investigation, an efficient recursive method is developed to solve the inverse heat transfer problem of estimating the wall heat flux on the wafer from the measurements of wafer temperature. The present recursive method is based on the Kalman filtering technique and the Karhunen–Loeve Galerkin procedure. Although the direct implementation of the Kalman filter on heat conduction equation is never feasible due to the tremendous requirement of computer time and memory, a practical method of recursive estimation is devised in the present investigation by reducing the partial differential equation to a minimal set of ordinary differential equations by means of the Karhunen–Loeve Galerkin procedure.

The Karhunen–Loeve Galerkin method, which is a type of Galerkin method that employs the empirical eigenfunctions of the Karhunen–Loeve decomposition as basis functions, is shown to solve inverse natural convection problems efficiently. The specific problem investigated is the inverse natural convection problem of determining the time-varying strength of a heat source from temperature measurement in the domain. The Karhunen–Loeve Galerkin procedure can reduce the Boussinesq equation to a set of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. The performance of the present technique of inverse analysis using the Karhunen–Loeve Galerkin procedure is assessed in comparison with the traditional technique of employing the Boussinesq equation, and is found to be very accurate as well as efficient.

The inverse natural convection problem of determining heat flux at the bottom wall of a two-dimensional cavity from temperature measurement in the domain is investigated by means of the Karhunen–Loeve Galerkin procedure. The Karhunen–Loeve Galerkin procedure, which is a type of Galerkin method that employs the empirical eigenfunctions of the Karhunen–Loeve decomposition as basis functions, can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. Previously, it had been demonstrated that the problems of optimal control of Burgers equation [H.M. Park, M.W. Lee, Y.D. Jang, Comput. Methods Appl. Mech. Engrg. 166 (1998) 289–308] and the Navier–Stokes equation [H.M. Park, M.W. Lee, Comput. Methods Appl. Mech. Engrg., 1999 (in press)] can be solved very efficiently through the reduction of modes based on the Karhunen–Loeve Galerkin procedure. In the present investigation, this technique is applied to the solution of inverse natural convection problem of estimating unknown wall heat flux. The performance of the present technique of inverse analysis using the Karhunen–Loeve Galerkin procedure is assessed in comparison with a traditional technique employing the Boussinesq equation, and is found to be very accurate as well as efficient.

A recursive algorithm is developed to solve the inverse heat conduction problem of estimating time-varying strength of a heat source from the knowledge of temperature readings taken inside the domain. It is based on the Kalman filtering technique. Although the recursive nature of the Kalman filter algorithm allows the possibility of on-line estimation in place of batch form off-line estimation, the straightforward implementation of this algorithm to multidimensional partial differential equations has not been feasible due to the tremendous requirement of computer time and memory. In the present investigation we overcome this difficulty by employing the Karbunen–Loeve Galerkin procedure which reduces the governing partial differential equation to a minimum number of ordinary differential equations. The performance of the present technique of inverse analysis is evaluated by several numerical experiments, and is found to be very accurate as well as efficient.

The Karhunen–Loeve Galerkin procedure is a type of Galerkin methods that employs the empirical eigenfunctions of the Karhunen–Loeve decomposition as basis functions. This technique can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. Previously [1] , it has been shown that one dimensional Burgers equation is reduced to a low dimensional model by this method, which is employed to solve boundary optimal control problems very efficiently. The present paper demonstrates that the Karhunen–Loeve Galerkin procedure can be extended to solve problems of the boundary optimal control of multidimensional Navier–Stokes equations. Since the reduction of modes in the multidimensional case is much larger than that in the one dimensional case, the present technique is found to be more powerful when applied to the control problems of the Navier–Stokes equation than those of the Burgers equation.