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Chapter 7 Boundary Conditions

By Allen Burns,2014-05-07 17:11
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Chapter 7 Boundary Conditions

Chapter 8: Boundary Conditions

     If we have a general flow field in the ocean, it will come as no surprise that the

    flow fields are dramatically different for a submarine or a turtle moving through the same

    medium. The equations we derived in chapter 6 make no discrimination of the object

    under consideration so there must be a significant contribution to the overall flow field

    defined at the various boundaries of the problem. By “boundary” we mean both the

    surfaces of any object contained in the fluid as well as the “container” holding the fluid under consideration. For the ocean, the container is specified by the surface, the ocean

    bottom and the surrounding continents. For the atmosphere, the container usually

    consists of the rarefied upper layers of the atmosphere and the planetary surface. To

    avoid any complication, it is assumed in the following analysis that the materials in

    contact with each other are at equilibrium and static.

Continuity of normal and tangential velocity at an interface:

    Consider an arbitrary fluid interface and examine a small right cylinder of height

    ^^

     and areas and that cuts through this boundary as seen in figure 1. ?An?And?1212

    ^

     ?An11medium 1

    ^

     t

    medium 2 d?

    ^

    ?An 22

    Figure 1 Infinitesimal cylinder with height in-between two separate mediums. d?

    Now apply conservation of mass over the cylinder boundaries under the limit

    . Under the limit, the density field will not vary considerably and thus we can d??0

    ??u?0apply the Boussinesq condition and approximate the continuity condition as .

    Integrating over the cylinder volume we see that

    ^^

     lim??udV?limu?dA?u?n?A?u?n?A?0121122??V?A00d??d??

Without loss of generality, we can assume that the cylinder is a right cylinder and

    ^^^

    ?A??A and . At the fluid interface we obtain the requirement of normal n?n??n1212

    velocity at the interface:

    ^^

     (1) u?n?u?n12

     Equation (1) should make sense to the reader, since if the normal components of

    the velocity field were not continuous, the resulting discontinuity would essentially be a

    rupture or shock in the medium and we have not assumed any event that would cause

    such a phenomena. Further, any discontinuity along a boundary would result in a

    transport of the differences of a given property amongst the two media; ultimately

    leading to a return to equilibrium and continuity. It is on this basis of physical reasoning

    that we can safely assume that, provided the mediums under observation are close to

    equilibrium, velocity and momentum is conserved at that interface. Conservation of

    tangential velocity is also called the no-slip condition and is expressed as:

    ^^

     (2) u?t?u?t12

     A unique circumstance of this no-slip condition is when we examine the interface

    between a fluid as our first medium and stationary solid object as our second medium. In

    ^^

    this case, u?t?0, and by equation (2) we can see that u?t?0 along the surface. We 21

    can see how this can lead to major shears in the flow field when they are moving with

    significant relative speeds compared to any solid object immersed in the fluid.

(Dis)continuity of the stress tensor:

    The continuity condition for the stress tensor follows the same reasoning as the

    previous section. Any discontinuities of the momentum of molecular motion will be

    compensated by a net transport of momentum until equilibrium and continuity is

    achieved. Let us examine the normal component of the stress tensor in detail. Consider

    the static, non-rotating form of the Navier-Stokes equation in the following index

    notation format.

    ???u?1iji u?g?ji???x?xjooj

Integrating the above equation over the infinitesimal cylinder shown in figure 1 and

    applying Gauss Theorem, we obtain: (The superscripts in the following equation are used to discriminate between the mediums where they apply),

    ^^???u?1??12i??limugdVlim?ndA?ndA????? ijijjj??ji????V?Ad??0d??0?x????joo??

    gDue to continuity in the velocity field and by the definition of being the sum of all i

    conservative body forces, all terms on the left hand side of the above equation approach

    zero under the limit of . The only way the right hand side of the above integral d??0

approaches zero is if the integrand on the right hand side approaches zero for every point

    ^^

    ?'?n??''?n. On a molecular level, along the interface and we would expect jjijij

    however, we have neglected to consider the effects of surface tension in the above

    analysis.

    Any differences in the attractive potential forces between the molecules in two separate mediums will show up as an additional surface force at the interface;

    proportional to the curvature of that interface in a plane defined by the direction of

    applied stress forces. When this additional stress force is considered in the above

    analysis, the resulting boundary condition for the normal stress at an interface has the

    form:

    ^^^??1112?? (3) ??????n?n?nijijjji??RR12??

RR and are the principal radii of curvature of the fluid interface in a local surface 21

    ^plane with normal and is a surface tension coefficient. n?i

     The above effects of surface tension leading to a discontinuity in the stress field

    only applies in the normal direction since the attractive forces on the molecules between

    the two mediums will always apply in the direction normal to the fluid interface. Since

    no such difference in potential forces exists in the tangential direction, we can safely

    assume that the tangential component of the stress tensor is continuous in equilibrium

    ^^12??t???t?0 (4) ijijjj

     Equations (1) (4) make up most of the boundary conditions that we will need to

    consider for the physical systems presented in the rest of this course. It should be noted

    that the above analysis for stress tensors and velocity fields are just as applicable to

    temperature or salinity gradients and conservation of heat or (salinity) results in boundary

    conditions such as

    ^^

    kn??Q?kn??Q 1122

    Where k is the associated diffusion coefficient for the conserved quantity and the subscripts indicate the associated medium where the quantity is being measured.

In summary:

    We can preserve any differences in the flow field due to a turtle swimming in the ocean

    or a submarine traveling in the same waters provide we include the conditions defined by

    equations (1)-(4) in this section. These resulting boundary conditions were derived based

    on the assumption of equilibrium. In other words, any deviation of continuity in the field

    will result in a net transport causing a return to equilibrium. The only exception to this

    physical process is when we consider the stresses normal to the fluid interface where we

    must also take into account the surface tension due to the difference in attractive forces between the molecules in the two mediums.

     In application, the above derived equations will be used to deduce or measure some of the properties along the physical system boundaries. For example, in an ocean basin we know the ocean floor and continental boundaries do not move (for the most part) so we can assume that both the normal and tangential flow is 0 along those surfaces. Alternatively, we collect data from all over the globe to obtain an assessment of the surface wind field for forecasting data. Further, from a mathematical standpoint, the application of boundary conditions ensures that the solutions to our problems are self-consistent. The equations of motion, conservation of mass and the equation of state allow us to solve for the velocity field, pressure and density as a well posed problem. In order for the solution to be unique, we need an appropriate set of boundary conditions

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