Averaged Continuity Equation

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Bruce R Locke - One of the best experts on this subject based on the ideXlab platform.

  • effects of spatial variation of cells and nutrient and product concentrations coupled with product inhibition on cell growth in a polymer scaffold
    Biotechnology and Bioengineering, 1999
    Co-Authors: Craig J. Galbán, Bruce R Locke
    Abstract:

    The effects of spatial variation of cells and nutrient and product concentration, in combination with product inhibition in cell growth kinetics on chondrocyte generation in a polymer scaffold, are analyzed. Experimental studies reported previously have demonstrated spatial dependence in the cultivation of chondrocytes. In the present study, the cell–polymer system is assumed to consist of two distinct phases. The cells, fluid, polymer matrix, and extracellular matrix comprise one phase, and the other phase consists of a fluid and polymer matrix. The only two species in the fluid considered to affect cell growth are the nutrient and product. The multiphase transport process of these two species in the cell–polymer system is described by the species Continuity Equations and corresponding boundary conditions for each individual phase. A volume-averaging approach is utilized for this system to derive Averaged species Continuity Equations for the nutrient and product concentrations. The volume-averaging approach allows for a single species in a two-phase system to be represented by a single Averaged Continuity Equation. Competitive product inhibition, saturation kinetics of substrate, and cell population control are assumed to affect the cell growth kinetics. A modified Contois growth kinetic model is used to represent the three factors that affect cell growth. A parameter analysis is performed and the results are compared qualitatively with experimental data found in the literature. © 1999 John Wiley & Sons, Inc. Biotechnol Bioeng 64: 633–643, 1999.

  • Effects of spatial variation of cells and nutrient and product concentrations coupled with product inhibition on cell growth in a polymer scaffold
    Biotechnology and Bioengineering, 1999
    Co-Authors: Craig J. Galbán, Bruce R Locke
    Abstract:

    The effects of spatial variation of cells and nutrient and product concentration, in combination with product inhibition in cell growth kinetics on chondrocyte generation in a polymer scaffold, are analyzed. Experimental studies reported previously have demonstrated spatial dependence in the cultivation of chondrocytes. In the present study, the cell-polymer system is assumed to consist of two distinct phases. The cells, fluid, polymer matrix, and extracellular matrix comprise one phase, and the other phase consists of a fluid and polymer matrix. The only two species in the fluid considered to affect cell growth are the nutrient and product. The multiphase transport process of these two species in the cell-polymer system is described by the species Continuity Equations and corresponding boundary conditions for each individual phase. A volume-averaging approach is utilized for this system to derive Averaged species Continuity Equations for the nutrient and product concentrations. The volume-averaging approach allows for a single species in a two-phase system to be represented by a single Averaged Continuity Equation. Competitive product inhibition, saturation kinetics of substrate, and cell population control are assumed to affect the cell growth kinetics. A modified Contois growth kinetic model is used to represent the three factors that affect cell growth. A parameter analysis is performed and the results are compared qualitatively with experimental data found in the literature.

Craig J. Galbán - One of the best experts on this subject based on the ideXlab platform.

  • effects of spatial variation of cells and nutrient and product concentrations coupled with product inhibition on cell growth in a polymer scaffold
    Biotechnology and Bioengineering, 1999
    Co-Authors: Craig J. Galbán, Bruce R Locke
    Abstract:

    The effects of spatial variation of cells and nutrient and product concentration, in combination with product inhibition in cell growth kinetics on chondrocyte generation in a polymer scaffold, are analyzed. Experimental studies reported previously have demonstrated spatial dependence in the cultivation of chondrocytes. In the present study, the cell–polymer system is assumed to consist of two distinct phases. The cells, fluid, polymer matrix, and extracellular matrix comprise one phase, and the other phase consists of a fluid and polymer matrix. The only two species in the fluid considered to affect cell growth are the nutrient and product. The multiphase transport process of these two species in the cell–polymer system is described by the species Continuity Equations and corresponding boundary conditions for each individual phase. A volume-averaging approach is utilized for this system to derive Averaged species Continuity Equations for the nutrient and product concentrations. The volume-averaging approach allows for a single species in a two-phase system to be represented by a single Averaged Continuity Equation. Competitive product inhibition, saturation kinetics of substrate, and cell population control are assumed to affect the cell growth kinetics. A modified Contois growth kinetic model is used to represent the three factors that affect cell growth. A parameter analysis is performed and the results are compared qualitatively with experimental data found in the literature. © 1999 John Wiley & Sons, Inc. Biotechnol Bioeng 64: 633–643, 1999.

  • Effects of spatial variation of cells and nutrient and product concentrations coupled with product inhibition on cell growth in a polymer scaffold
    Biotechnology and Bioengineering, 1999
    Co-Authors: Craig J. Galbán, Bruce R Locke
    Abstract:

    The effects of spatial variation of cells and nutrient and product concentration, in combination with product inhibition in cell growth kinetics on chondrocyte generation in a polymer scaffold, are analyzed. Experimental studies reported previously have demonstrated spatial dependence in the cultivation of chondrocytes. In the present study, the cell-polymer system is assumed to consist of two distinct phases. The cells, fluid, polymer matrix, and extracellular matrix comprise one phase, and the other phase consists of a fluid and polymer matrix. The only two species in the fluid considered to affect cell growth are the nutrient and product. The multiphase transport process of these two species in the cell-polymer system is described by the species Continuity Equations and corresponding boundary conditions for each individual phase. A volume-averaging approach is utilized for this system to derive Averaged species Continuity Equations for the nutrient and product concentrations. The volume-averaging approach allows for a single species in a two-phase system to be represented by a single Averaged Continuity Equation. Competitive product inhibition, saturation kinetics of substrate, and cell population control are assumed to affect the cell growth kinetics. A modified Contois growth kinetic model is used to represent the three factors that affect cell growth. A parameter analysis is performed and the results are compared qualitatively with experimental data found in the literature.

Subhasish Dey - One of the best experts on this subject based on the ideXlab platform.

  • Turbulence in Open-Channel Flows
    GeoPlanet: Earth and Planetary Sciences, 2014
    Co-Authors: Subhasish Dey
    Abstract:

    The turbulence in a fluid flow is characterized by irregular and chaotic motion of fluid particles. It is a complex phenomenon. In this chapter, the turbulence characteristics are discussed with reference to flow over a sediment bed. An application of Reynolds decomposition and time-averaging to the Navier–Stokes Equations yields the Reynolds-Averaged Navier–Stokes (RANS) Equations, containing terms of Reynolds stresses. The RANS Equations along with the time-Averaged Continuity Equation are the main Equations to analyze turbulent flow. The classical turbulence theories were proposed by Prandtl and von Karman. Prandtl simulated the momentum exchange on a macro-scale to explain the mixing phenomenon in a turbulent flow establishing the mixing length theory, while von Karman’s relationship for the mixing length is based on the similarity hypothesis. The velocity distribution in open-channel flow follows the linear law in viscous sublayer, the logarithmic law in turbulent wall shear layer, and the wake law in the outer layer. The determination of bed shear stress is always a challenging task. Different methods for the determination of bed shear stress are discussed. Flow in a narrow channel exhibits strong turbulence-induced secondary currents, and as a result, the maximum velocity appears below the free surface, known as dip phenomenon. Isotropic turbulence theory deals with the turbulent kinetic energy (TKE) transfer from the large-scale motions to smaller-scale motions until attaining an adequately small length scale so that the fluid molecular viscosity can dissipate the TKE into heat. Anisotropy in turbulence is analyzed by the anisotropic invariant mapping (AIM) and the anisotropy invariant function to quantify the degree of the departure from isotropy. Higher-order correlations are given by skewness and kurtosis of velocity fluctuations, TKE flux, and budget. This chapter also includes most of the modern development of turbulent phenomena, such as coherent structures and burst phenomena and double-averaging of heterogeneous flow over gravel beds.

  • Instability Theory of Sand Ripples Formed by Turbulent Shear Flows
    Journal of Hydraulic Engineering, 2012
    Co-Authors: Sujit K. Bose, Subhasish Dey
    Abstract:

    AbstractA theory of turbulent shear flow over a sand bed is developed, addressing the instability principle of the fluid-granular bed interface leading to the formation of ripples. The Reynolds-Averaged Navier-Stokes (RANS) Equations and the time-Averaged Continuity Equation are analyzed using a 1/7-power law of the time-Averaged streamwise velocity and treating the curvilinear streamlines by the Boussinesq approximation. The integration of the RANS Equations leads to a governing dynamical Equation of flow over a mobile bed. A near-bed flow layer of 3.5 times the ripple height is considered being affected by the ripples. The dynamical Equation of the mobile sand bed is based on the Exner’s sediment Continuity Equation in conjunction with the Meyer-Peter and Muller bed-load transport formula as modified to account for the effect of local bed slope attributable to bed forms. The coupled dynamical Equations are then analyzed to estimate the parameters for the instability that results in the formation of ripp...

  • Reynolds Averaged theory of turbulent shear flows over undulating beds and formation of sand waves.
    Physical review. E Statistical nonlinear and soft matter physics, 2009
    Co-Authors: Sujit K. Bose, Subhasish Dey
    Abstract:

    Based on the Reynolds Averaged Navier-Stokes (RANS) Equations and the time-Averaged Continuity Equation, a theory of turbulent shear flow over an undulating sand bed is developed addressing the instability criterion of plane sand beds in free-surface flows leading to the formation of sand waves. In the analysis, the integration of RANS Equations leads to generalized Saint Venant Equations, in which the time-Averaged streamwise velocity is characterized by a power law obtained from turbulence closure, treating the curvilinear streamlines by the Boussinesq approximation. As a consequence, the modified pressure distribution has a departure from the traditionally linear hydrostatic pressure distribution. The instability analysis of a plane sand bed yields the curves of the Froude number versus nondimensional wave number, determining an instability zone for which at lower Froude numbers (less than 0.8), the plane bed becomes unstable with the formation of dunes; whereas at higher Froude numbers, the plane bed becomes unstable with the formation of standing waves and antidunes. For higher Froude numbers, the experimental data for antidunes lie within the unstable zone; while for lower Froude numbers, the same is found for dunes with some experimental scatter.

Sujit K. Bose - One of the best experts on this subject based on the ideXlab platform.

  • Instability Theory of Sand Ripples Formed by Turbulent Shear Flows
    Journal of Hydraulic Engineering, 2012
    Co-Authors: Sujit K. Bose, Subhasish Dey
    Abstract:

    AbstractA theory of turbulent shear flow over a sand bed is developed, addressing the instability principle of the fluid-granular bed interface leading to the formation of ripples. The Reynolds-Averaged Navier-Stokes (RANS) Equations and the time-Averaged Continuity Equation are analyzed using a 1/7-power law of the time-Averaged streamwise velocity and treating the curvilinear streamlines by the Boussinesq approximation. The integration of the RANS Equations leads to a governing dynamical Equation of flow over a mobile bed. A near-bed flow layer of 3.5 times the ripple height is considered being affected by the ripples. The dynamical Equation of the mobile sand bed is based on the Exner’s sediment Continuity Equation in conjunction with the Meyer-Peter and Muller bed-load transport formula as modified to account for the effect of local bed slope attributable to bed forms. The coupled dynamical Equations are then analyzed to estimate the parameters for the instability that results in the formation of ripp...

  • Reynolds Averaged theory of turbulent shear flows over undulating beds and formation of sand waves.
    Physical review. E Statistical nonlinear and soft matter physics, 2009
    Co-Authors: Sujit K. Bose, Subhasish Dey
    Abstract:

    Based on the Reynolds Averaged Navier-Stokes (RANS) Equations and the time-Averaged Continuity Equation, a theory of turbulent shear flow over an undulating sand bed is developed addressing the instability criterion of plane sand beds in free-surface flows leading to the formation of sand waves. In the analysis, the integration of RANS Equations leads to generalized Saint Venant Equations, in which the time-Averaged streamwise velocity is characterized by a power law obtained from turbulence closure, treating the curvilinear streamlines by the Boussinesq approximation. As a consequence, the modified pressure distribution has a departure from the traditionally linear hydrostatic pressure distribution. The instability analysis of a plane sand bed yields the curves of the Froude number versus nondimensional wave number, determining an instability zone for which at lower Froude numbers (less than 0.8), the plane bed becomes unstable with the formation of dunes; whereas at higher Froude numbers, the plane bed becomes unstable with the formation of standing waves and antidunes. For higher Froude numbers, the experimental data for antidunes lie within the unstable zone; while for lower Froude numbers, the same is found for dunes with some experimental scatter.

Sergio Fagherazzi - One of the best experts on this subject based on the ideXlab platform.

  • Basic flow field in a tidal basin
    Geophysical Research Letters, 2002
    Co-Authors: Sergio Fagherazzi
    Abstract:

    [1] A simplified model for tidal flow in a basin is presented. The model is based on the assumption of a flat water surface oscillating synchronously in the tidal basin. Under this hypothesis the depth-Averaged Continuity Equation becomes a Poisson Equation that can be easily resolved at each instant of the tidal cycle. This formulation, which is particularly valid for small, deep basins, provides a simplified solution of the depth-integrated shallow water Equations and suggests a possible approach to model long-term morphodynamic evolution of tidal basins. The model is tested in San Diego Bay, California, and the results are briefly discussed.