Physical Processes Georg Umgiesser and Ramiro Neves CONTENTS 3.1 Introduction to Transport Phenomena 3.2 Fluxes and Transport Equation 3.2.1 Velocity and Diffusivity in Laminar and Turbulent Flows and in a Numerical Model 3.2.2 Advective Flux 3.2.3 Diffusive Flux 3.2.4 Elementary Area and Elementary Volume 3.2.5 Net Flux across a Closed Surface 3.3 Transport and Evolution 3.3.1 Rate of Accumulation 3.3.2 Lagrangian Form of the Evolution Equation 3.3.3 Eulerian Form of the Evolution Equation 3.3.4 Differential Form of the Transport Equation 3.3.5 Boundary and Initial Conditions 3.4 Hydrodynamics 3.4.1 Conservation Laws in Hydrodynamics 3.4.1.1 Conservation of Mass 3.4.1.2 Conservation of Momentum 3.4.1.2.1 The Euler Equations 3.4.1.2.2 The Euler Equations in a Rotating Frame or Reference 3.4.1.2.3 The Navier-Stokes Equations 3.4.1.3 Conservation of Energy 3.4.1.4 Conservation of Salt 3.4.1.5 Equation of State 3.4.2 Simplification and Scale Analysis 3.4.2.1 Incompressibility 3.4.2.2 The Hydrostatic Approximation 3.4.2.3 The Coriolis Force 3.4.2.4 The Reynolds Equations 3.4.2.5 The Primitive Equations 3.4.3 Special Flows and Simplifications in Dimensionality 3.4.3.1 Barotrophic 2D Equations 3.4.3.2 1D Equations (Channel Flow) 3 L1686_C03.fm Page 43 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press 3.4.4 Initial and Boundary Conditions 3.4.4.1 Initial Conditions 3.4.4.2 Conditions on Material Boundaries 3.4.4.3 Conditions on Open Boundaries 3.4.4.4 Conditions on the Sea Surface and the Sea Bottom 3.5 Boundary Processes 3.5.1 Bottom Processes 3.5.1.1 Bottom Shear Stress 3.5.1.2 Other Bottom Processes 3.5.2 Solid Boundary Processes 3.5.3 Free Surface Processes 3.5.3.1 Mass Exchange 3.5.3.2 Momentum Exchange 3.5.3.3 Energy Exchange 3.5.4 Cohesive and Noncohesive Sediment Processes Bibliography 3.1 INTRODUCTION TO TRANSPORT PHENOMENA Chapter 2 concluded that the calculation of spatio-temporal distribution of major components of a lagoon’s hydrogeomorphological unit and biocoenose is important for the description of the structure and function dynamics (productivity and carrying capacity) of the lagoon system and, consequently, for sustainable management. This concept is required to understand the transport phenomena that describe the evolution of properties due to fluid motion (advection) and/or molecular and turbulent dynam- ics (diffusion). In the case of turbulent flows the small-scale motion of the fluid particles is actually random, and this nonresolved advection is also treated as diffu- sion (eddy diffusion). A mathematical description of the transport phenomena (transport equations) is based on the concept of conservation principle, which is valid in any application. Conservation principle can be stated as {The rate of accumulation of a property inside a control volume} = {what flows in minus what flows out} + {production minus consumption} Using this conservation principle, transport equations for any property inside a control volume can be derived if production and consumption mechanisms are known and if the control volume and transport processes are quantified. The control volume is presented as the largest volume for which one can consider the interior properties as uniformly distributed as well as fluxes across the surface. In previous coastal lagoon studies the control volume was often implicitly defined as the whole lagoon. Concepts of residence and flushing time were derived from this global approach (see Chapter 5 for details). In that case, only fluxes at the boundaries were required. This type of integral approach cannot describe gradients and is consequently not sufficient to support process-oriented research L1686_C03.fm Page 44 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press or management. For these purposes the system has to be divided into homogeneous parts (control volumes) and fluxes between them have to be calculated. This approach requires a numerical model. The number of space dimensions required to describe the control volumes equals the number of dimensions of the model. The resolution of the transport equations in practical situations has been made possible using numerical methods and computers (see Chapter 6 for details). Before the advent of computers transport processes had to be studied using empir- ical formulations (derived from experiments) or analytical solutions in simple geometries or boundary conditions. This chapter presents a general transport equation (also called an evolution equa- tion) based on the concepts of (1) control volume, (2) advective flux, and (3) diffusive flux. Based on this generic equation, equations for hydrodynamics, temperature, salin- ity, and suspended sediments are also introduced. Special flows and simplification of dimensionality and boundary processes and conditions, particularly for coastal lagoons, are described in detail for use in lagoon modeling studies. 3.2 FLUXES AND TRANSPORT EQUATION 3.2.1 V ELOCITY AND D IFFUSIVITY IN L AMINAR AND T URBULENT F LOWS AND IN A N UMERICAL M ODEL For transport purposes, fluids are considered a continuum system. Velocity is defined in a macroscopic way based on the concept of continuum system. Because fluids are not a real continuum, system velocity cannot describe transport processes at the molecular scale. The nonrepresented processes are represented by diffusion. Although the concept of velocity is well known, it is reconsidered for modeling purposes. Diffusion in laminar flows occurs from movements at a molecular scale not represented by the velocity. In turbulent flows, velocity, as defined for laminar flows, becomes time dependent, changing at a frequency that is too high to be represented analytically. As a consequence time average values must be considered, following the Reynolds approach (see Section 3.4.2.4 for details). Transport processes, not described by this average velocity, are represented by turbulent diffusion (using an eddy diffusivity, which is several orders of magnitude higher than molecular diffu- sivity). More information on this topic is given in Section 3.4. Most numerical models use grids with spatial and time steps larger than those associated with turbulent eddies. Again, processes not resolved by velocity computed by models have to be accounted for by diffusion (subgrid diffusion). The box represented in Figure 3.1 is commonly used to illustrate the concept of diffusion in laminar flows. The same box could be used to illustrate the concept of eddy diffusion in turbulent flows or subgrid diffusion in numerical models. In molecular diffusion white and black dots represent molecules, while in other cases they represent eddies. In the initial conditions (stage (a) in Figure 3.1) two different fluids are kept apart by a diaphragm. Molecules inside each half-box move randomly, with velocities not described by our model (Brownian or eddy). When the diaphragm is removed, particles from each side keep moving, resulting in the possibility of L1686_C03.fm Page 45 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press mixing (stage (b) in Figure 3.1). After some time the proportion of white and dark fluid is the same in both the box halves. At this stage, the probability of a white molecule moving from the right side of the box to the left side is equal to the probability of another white molecule moving the opposite way and, therefore, there is no net exchange (stage (c) in Figure 3.1). A macroscopic view of the mixing is represented in Figure 3.2. In the initial condition (stage (a) in Figure 3.2), a black and a white side is observed (stage (b) in Figure 3.2); during the mixing process a growing gray area occurs, corresponding to the mixing zone; and after complete mixing, a homogeneous gray fluid is observed (stage (c) in Figure 3.2). The velocity of each elementary portion of fluid, like the velocity of any other material point, is defined using two consecutive locations, as shown in Figure 3.3: (3.1) Knowing the velocity of each individual molecule, it will be possible to fully characterize transport. But, according to Heisenberg’s uncertainty principle, it will never be possible to know the place and the velocity of each molecule simultaneously. Consequently, the fluid has to be considered a continuum system, for which a velocity is defined. FIGURE 3.1 Distribution of molecules of two different substances, inside a box at three different moments: initially kept apart by a diaphragm (a), during mixing (b), and when spatial gradient has disappeared (c). FIGURE 3.2 Macroscopic view of the fluid composed by the molecules represented in Figure 3.1. ( a ) ( b ) ( c ) v rr r u dx x t dx dt dx dt u t tt t i i = − == = → + lim ∆ ∆ ∆ 0 (a) (b) (c) L1686_C03.fm Page 46 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press Considering a fluid as a continuum system, an elementary volume of fluid is a portion of fluid so small that its properties (including velocity) can be considered as uniform, but it is much bigger than the size of a molecule in laminar flow and than an eddy in turbulent flow. In the case of a numerical model, an elementary volume is the volume included inside a grid cell. As a consequence, diffusivity increases with grid size in numerical models. 3.2.2 A DVECTIVE F LUX The advective flux accounts for the amount of a property transported per unit of time due to fluid velocity across a surface perpendicular to the motion. Its dimensions are [ BT − 1 ] and can be expressed as † (3.2) where V is the volume. The quantity β = B / V has the dimensions of a specific quantity (amount per unit of volume) and is called the concentration. The ratio between the volume and the time [ L 3 T − 1 ] is the flow rate that can be calculated as the product FIGURE 3.3 Trajectory of an elementary portion of fluid showing consecutive locations apart in time of ∆ t . † [ B ] means “dimensions of B ”; T means time. X 2 X 1 X 3 r x t r x t +∆ t Φ adv B V V t = L1686_C03.fm Page 47 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press of the velocity [ LT − 1 ] and the area [ L 2 ] of the surface. The transport produced by the velocity per unit of area is where is the velocity relative to the velocity of the surface. This flux is a vector parallel to the velocity. The flux across an elementary area dA not perpendicular to the velocity is given by where is the external normal to the elementary surface dA . In the case of a finite surface A , the total flux is the summation of the elementary fluxes across the elementary areas composing it. This summation can be represented by the integral over that surface: (3.3) This is the generic definition of the advective flux of a property B across a surface A . 3.2.3 D IFFUSIVE F LUX Diffusive flux is the net transport associated with the Brownian movement of mol- ecules in the case of a laminar flow or due to both molecular and turbulent movement in the case of a turbulent flow, depending on the property gradient. Diffusion transports the property down the direction of the gradient, as determined by Fick, who stated that the diffusive flux per unit of area is given by (3.4) where ϕ is the diffusivity and the quantity inside the parentheses is the gradient of β , the specific value of B (concentration in case of mass). Both β gradient and diffusivity can vary spatially, implying the calculation of the flux on elementary surfaces: and its integration along the overall surface: (3.5) where is the normal to the elementary surface. φβ adv u=⋅ r r u dundA adv ()()Φ= ⋅ β v r r n ΦΦ adv adv AA dundA==⋅ ∫∫ () () β rr r r φϕβ dif =− ∇() dndA dif () () φϕβ =− ∇ r r Φ dif dif AA dndA==−∇ ∫∫ () () φϕβ r r r n L1686_C03.fm Page 48 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press 3.2.4 E LEMENTARY A REA AND E LEMENTARY V OLUME The transport across a finite surface A is obtained as the integration of the transport across elementary areas where properties can be assumed to be uniform. Mathemati- cally, elementary areas are infinitesimal, but physically they are just small enough to allow the assumption that properties assume constant values on their surfaces, allowing for the substitution of the integrals by summations. Fluxes across an elementary surface are obtained by multiplying fluxes per unit of area with the area of the elementary surface. This is a basic assumption of modeling. An elementary volume is a volume limited by elementary areas, inside which properties can be considered as having uniform values. The total amount of a property contained inside an elementary volume is given by the product of its specific value with its elementary volume. 3.2.5 N ET F LUX ACROSS A C LOSED S URFACE Let us consider a closed surface as represented in Figure 3.4. In the figure an elementary area ∆ A , local normal n, and velocity u , which can vary from point to point, are represented. In regions where normal and velocity have opposite senses the internal product is negative, meaning that property B is being advected (trans- ported) into the interior of the volume limited by the surface. Where the internal product is positive, the property is being transported outward. Thus, the integral of the flux (advective or diffusive) over a closed surface gives the difference between the amount of property being transported outward and inward. 3.3 TRANSPORT AND EVOLUTION An evolution equation describes the transformations suffered by a property as time progresses. The properties of a fluid limited by solid surface can be modified only by production (sources) or destruction (sinks) processes. If the boundary of the FIGURE 3.4 Representation of a generic transparent surface, an elementary area on that surface, the respective exterior normal, and the fluid velocity. L1686_C03.fm Page 49 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press volume is permeable, allowing for advective and/or diffusive fluxes, transport also will contribute for the evolution of the property inside the volume. 3.3.1 R ATE OF A CCUMULATION The rate of accumulation of a property B inside an elementary volume is the ratio between the variation of the total amount contained in the volume and the time interval during which accumulation happened. This concept can be translated by the algebraic equation: In this equation, the total amount of B inside an elementary volume V in each moment is given by the product of the specific value β multiplied with the volume. † If an infinitesimal time interval is considered, the previous equation can be written in a differential form: This equation puts into evidence the physical meaning of the time derivative, showing that it describes the rate of accumulation. 3.3.2 LAGRANGIAN FORM OF THE EVOLUTION EQUATION If there is no flux across the boundary of the control volume, the rate of accumulation accounts for the sources (S 0 ) minus the sinks (S i ): This can happen in the case of a volume limited by a solid boundary or in the case of a volume moving at the same velocity as the flow where diffusivity can be neglected. Faecal bacteria are traditionally assumed to be unable to grow in saline water and have a first-order decay rate. For such a variable in a volume where advective and diffusive fluxes are null and the sink is the mortality of bacteria, the evolution equation would be written as where m is the rate of mortality. † In a general case, the total amount inside a volume V of a property with a specific value β is the integral of β inside the volume. In the case that β is constant, the integral is the product of β times the volume. () () ββ VV t tt t+ − ∆ ∆ dV dt () β dV dt VS S oi () () β =− dV dt mV () β β =− L1686_C03.fm Page 50 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press If the surface limiting the volume is permeable, a diffusive flux exists propor- tional to the gradient and the evolution equation becomes (3.6) Please note that a positive flux is directed out of the elementary volume and is, therefore, lowering the mass of the property inside the control volume. The negative sign in front of the integral accounts for this fact. Equation (3.6) describes the evolution of a generic property B, with a specific value β , if there is no advective exchange between the elementary volume of fluid and the surrounding volume. That is the case when the elementary volume moves at the same velocity as the fluid (null relative velocity), corresponding to the Lagrangian formulation of the problem. Measuring instruments moving freely, trans- ported by the flow (e.g., attached to a buoy), would carry out this type of measure- ment. That is not, however, the common way of measuring. 3.3.3 EULERIAN FORM OF THE EVOLUTION EQUATION In general, field measurements are carried out in fixed stations (see Chapter 7 for details). If such a monitoring strategy is adopted, the elementary volume to be considered in the evolution equation is fixed in space, and the velocity relative to the surface of the volume becomes the flow velocity. In that case the evolution equation becomes: (3.7) The partial derivative states that the control volume does not move and, consequently, the velocity to be considered in the advective flux is the flow velocity. This equation holds for situations where the volume is time dependent. If we consider the case of a rigid volume, we can take it out of the time derivative. 3.3.4 DIFFERENTIAL FORM OF THE TRANSPORT EQUATION Let us consider a Cartesian reference and an elementary cubic volume, as represented in Figure 3.5. Being an elementary volume, it is small enough to assume that properties have uniform values on the surface and that the value of the property can be considered uniform inside the volume. Considering this approach and the geo- metric properties indicated in the figure, we can write an equation for the volume and for the fluxes. The volume is given by ∆V = ∆x 1 ∆x 2 ∆x 3 The integral of fluxes becomes the summation of their specific values across each elementary area (control volume surfaces) multiplied by the area of the dV dt ndA V S S i A () () ( ) β ϕβ =− − ∇ + − ∫ r r 0 dV dt un n dA V S S i A () [()]() β βϕβ =− − ∇ + − ∫ rr r r 0 L1686_C03.fm Page 51 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press corresponding surface. Keeping the orientation of the normal in mind, we can write the fluxes for surfaces perpendicular to the x 1 axis: Doing a similar calculation for other directions, dividing the whole equation by ∆V and letting ∆x i converge to zero, we obtain Equation (3.8): (3.8) In this equation the Einstein convention † for summation is used and sources and sinks are calculated per unit of volume. Equation (3.7) and Equation (3.8) are general and hold for any property. They can also be used to derive continuity and momentum equations in the next sections. FIGURE 3.5 Cubic type elementary control volume. Velocity components are represented on every surface of the volume. † In the Einstein convention, a doubled index represents a summation of the terms obtained replacing that index by each dimension of the physical space (3 in a three-dimensional space). x 2 x 1 x 3 (u 1 ) x 1 (u 1 ) x 1 +∆x 1 (u 2 ) x 2 +∆x 2 (u 2 ) x 2 ∆ x 1 ∆ x 2 ∆ x 3 βϕ φ βϕ φ u x u x xx xx x 1 1 1 1 23 11 1 − ∂ ∂ −− ∂ ∂ +∆ ∆∆ ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ +− β β ϕ φ t u xxx SS i ijj oi () L1686_C03.fm Page 52 Monday, November 1, 2004 3:33 PM © 2005 by CRC Press [...]... ∂x2 ∂x3 ∂x1 and the viscous forces in the other directions read F2r = ν ∂2u2 ∂x 2 j F3r = ν ∂2u3 ∂x 2 j Including the friction term into the Euler equations we end up with the so-called Navier-Stokes equations: ∂ 2u du1 1 ∂p − fu2 = − + ν 21 ∂x j dt ρ ∂x1 du2 ∂ 2u 1 ∂p + fu1 = − + ν 22 dt ρ ∂x2 ∂x j (3. 15b) du3 ∂ 2u 1 ∂p =− − g + ν 23 ρ ∂x3 dt ∂x j © 2005 by CRC Press (3. 15a) (3. 15c) L1686_C 03. fm... L1686_C 03. fm Page 66 Monday, November 1, 2004 3: 33 PM The conservation of mass leads to the continuity equation ∂u1 ∂u2 ∂u3 + + =0 ∂x1 ∂x2 ∂x3 (3. 21) The two horizontal components of the Reynolds equations read 2 ∂u1 ∂u ∂u ∂u ∂2 u V ∂2 u 1 ∂p H ∂ u + u1 1 + u2 1 + u3 1 − fu2 = − + vM 21 + 21 + vM 21 ∂t ∂x1 ∂x2 ∂x3 ρ ∂x1 ∂x2 ∂x3 ∂x1 (3. 22a) and 2 ∂u2 ∂u ∂u ∂u ∂2 u V ∂2 u2 1 ∂p H ∂ u (3. 22b)...L1686_C 03. fm Page 53 Monday, November 1, 2004 3: 33 PM Following a similar procedure and assuming an incompressible fluid—an equation equivalent to Equation (3. 8) on a Lagrangian reference can be obtained from Equation (3. 6): dβ ∂ = dt ∂x j ∂β ϕ ∂x + ( So − Si ) j (3. 9) Comparing Equation (3. 8) and Equation (3. 9) and using the fact that for an incompressible fluid ∂ui/∂xi = 0 (see Section 3. 4.1.1),... parcel are composed not only of the pressure gradient, but © 2005 by CRC Press L1686_C 03. fm Page 58 Monday, November 1, 2004 3: 33 PM also of the additional Coriolis force, and the new equations (not deduced) can be written as du1 1 ∂p − fu2 = − dt ρ ∂x1 (3. 14a) du2 1 ∂p + fu1 = − dt ρ ∂x2 (3. 14b) du3 1 ∂p =− −g ρ ∂x3 dt (3. 14c) where f is the Coriolis parameter with f = 2Ω sin(ϕ), Ω is the angular frequency... (3. 23) The conservation equations for heat and salt read 2 ∂T ∂T ∂T ∂T ∂ 2T V ∂ 2T 1 ∂Qs H ∂ T + u1 + u2 + u3 = vT 2 + 2 + vT 2 + ∂t ∂x1 ∂x2 ∂x3 ∂x3 ρc p ∂x3 ∂x1 ∂x2 (3. 24a) and 2 ∂S ∂S ∂S ∂S ∂ 2S V ∂ 2S H ∂ S + u1 + u2 + u3 = vS 2 + 2 + vS 2 ∂t ∂x1 ∂x2 ∂x3 ∂x3 ∂x1 ∂x2 (3. 24b) with VT and VS the turbulent diffusivities for temperature and salinity, respectively Finally, the equation... Press L1686_C 03. fm Page 67 Monday, November 1, 2004 3: 33 PM 3. 4 .3 SPECIAL FLOWS AND SIMPLIFICATIONS IN DIMENSIONALITY The equations derived above explain the whole spectrum of dynamic behavior that may be expected in the coastal seas and lagoons Quite often, however, the flow shows some peculiar characteristics that make it possible to simplify the equations even more In this section some often-made simplifications... equivalent to the incorporation of the pressure gradient force r into the Euler equations, and the result for the viscous force F31 per unit mass in the x1 direction due to the shear in the x3 direction is r F31 = ∂2u1 1 ∂τ 31 1 ∂ ∂u1 = µ ∂x = ν ∂x 2 ρ ∂x3 ρ ∂x3 3 3 where we have used the kinematic viscosity coefficient v = µ/ρ to write the equation more compactly In the same way, the viscous... is the Coriolis force that is responsible for all the meso-scale structure we can see on the weather charts that contain cyclones and anticyclones However, the Coriolis force is important only for large-scale circulations and, therefore, may not be important for many coastal lagoons of the world 3. 4.1.2 .3 The Navier-Stokes Equations The above-derived Euler equations describe the flow of a fluid without... =0 ∂xi (3. 19) In this version the continuity equation states that the divergence of the mass flow is zero This is the form of the mass conservation normally used in oceanography 3. 4.2.2 The Hydrostatic Approximation If we consider a basin of water at rest (ui = 0), the Navier-Stokes equations reduce to 0=− © 2005 by CRC Press 1 ∂p + gi ρ ∂xi L1686_C 03. fm Page 63 Monday, November 1, 2004 3: 33 PM The... 2005 by CRC Press ∂η ∂BHu1 + =0 ∂t ∂x1 (3. 29) L1686_C 03. fm Page 70 Monday, November 1, 2004 3: 33 PM where B now denotes the (possibly variable) width of the channel The momentum equation now reads 2 ∂u1 ∂u 1 ∂η H ∂ u + + u1 1 = − g τ 1s − τ 1b + vM 21 ∂x1 ∂t ∂x1 ∂x1 ρ0 H ( ) (3. 30) and the other conservation equations for T and S can be deduced in a similar way 3. 4.4 INITIAL AND BOUNDARY CONDITIONS Boundary . Euler equations we end up with the so-called Navier-Stokes equations: (3. 15a) (3. 15b) (3. 15c) τµ 31 1 3 = ∂ ∂ u x F r 3 1 F xx u x u x r 31 31 33 1 3 2 1 3 2 11 = ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ρ τ ρ µν F u x u x u x u x r j 1 2 1 1 2 2 1 2 2 2 1 3 2 2 1 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ νν F u x F u x r j r j 2 2 2 2 3 2 3 2 = ∂ ∂ = ∂ ∂ νν du dt fu p x u x j 1 2 1 2 1 2 1 −=− ∂ ∂ + ∂ ∂ ρ ν du dt fu p x u x j 2 1 2 2 2 2 1 +=− ∂ ∂ + ∂ ∂ ρ ν du dt p x g u x j 3 3 2 3 2 1 =− ∂ ∂ −+ ∂ ∂ ρ ν L1686_C 03. fm. Equations 3. 4 .3 Special Flows and Simplifications in Dimensionality 3. 4 .3. 1 Barotrophic 2D Equations 3. 4 .3. 2 1D Equations (Channel Flow) 3 L1686_C 03. fm Page 43 Monday, November 1, 2004 3: 33 PM ©. Flux 3. 2.4 Elementary Area and Elementary Volume 3. 2.5 Net Flux across a Closed Surface 3. 3 Transport and Evolution 3. 3.1 Rate of Accumulation 3. 3.2 Lagrangian Form of the Evolution Equation 3. 3.3