2.4 Theory of turbulence
2.4.3 Vorticity dynamics
flow.
3. ~ω(∇.~u) =ωi∂ui
∂xj describes the effect ofexpansion or compression on the vorticity field. In a flow that is expanding, this term is positive, and results in a decrease in the magnitude of vorticity due to the minus sign in front of this term. If the fluid is undergoing compression, the magnitude of the vorticity will increase. In incompressible flows, this term will not play a role, but is however a very important mechanism in combustion and non-reacting compressible flows.
4. ~u(∇.~ω) =ui∂ωi
∂xj is always zero.
5. −∇ ×(1ρ∇p) which when expanded gives ρ12(∇ρ× ∇p) is called thebarometric torque and results in a generation of vorticity from unequal acceleration as a result of non-aligned density and pressure gradients.
In a uniform, constant density flow, this term is zero. Suppose the pressure gradient is perpendicular to the density gradient (the case for which barometric torque is largest). The lighter density fluid will be accelerated faster than the high density fluid, resulting in a shear layer, thus generating vorticity.
6. ν∇2ω describes the effects of viscous diffusion on the distribution of vorticity. As a result of viscosity, vorticity in the flow tends to diffuse in space. In high Reynolds number turbulent flows, viscous diffusion of vorticity will be dominated by the other mechanisms in the vorticity transport equations. This will be the case unless the length scales of the turbulence are small enough so that contributions of viscosity can be important. The effects of viscocity on the large scale vortex structure in a turbulent flow are generally small.
For constant density, we can write in cartesian form :
∂ωi
∂t +uj
∂ωi
∂xj −ωj
∂ui
∂xj
=ν ∂2ωi
∂xj∂xj
(2.89)
If we consider 2-D constant density flow, the vorticity equation reduces to :
∂ωi
∂t +uj∂ωi
∂xj
=ν ∂2ωi
∂xj∂xj
(2.90)
Under this restriction, the vorticity simply acts as a passive scalar that follows fluid particle paths so becomes a good indicator of fluid flow patterns. The vorticity vector is therefore confined to a plane perpendicular to the flow, and no enhancement of vorticity of transport to smaller scales by vortex stretching is possible. In this case vorticity just satisfies a standard convection-diffusion equation.
Chapter 3: Experimental facility and setup
”Thought experiment is in any case a necessary precondition for physical experiment. Every experi- menter and inventor must have the planned arrangement in his head before translating it into fact”.
-Ernst Mach (1838 - 1916)
3.1 Experimental facility
Experiments were performed in a wave tank/flume with a sloping beach at one end. The flume is 20 m long, 0.75 m wide and 0.8 m deep. For visualization purposes, the side walls of the flume are composed of tempered glass. Waves were generated by a servo-controlled piston type wave maker that has a maximum paddle stroke of 0.8 m and is designed for water depths of up to 0.75 m. Complementary documentation on the specifications of the wave maker and wave generation mechanism is available at theurl indicated by [116]. The flume was also fitted with a cemented beach having a slope of 1:20 (height : length). A 1:20 slope beach was chosen in order to get a long enough surf zone length over which measurements of wave parameters could later be conducted. Choosing a higher slope would result in a very short surf zone, alternatively choosing a smaller slope would result in waves breaking too close to the end of the flume, also resulting in a shorter surf zone. The 1:20 beach slope has also been used as the standard slope by numerous researchers in similar studies (Sou &Yeh [12];Nadaoka &Kondoh [37];Cowen et al.[60];
Lara et al.[117] ;Huang et al.[115];Sou et al.[75];Sakai et al.[118];Govender et al.[73]). Melville [119]
gives an extensive review of the different breaking wave geometries, wave breaking mechanisms, and evolution of the wave profile for deep water waves.
Figure 3.1 is a picture showing an aerial view of the laboratory flume. The wave maker is located near the bottom left corner. The above facility is located in the Coastal and Hydraulics Engineering Laboratory at the Council for Scientific and Industrial Research (CSIR) in Stellenbosch, South Africa.
Figure 3.2 shows the schematic view of the flume illustrating characteristic regions of the flume and giving overall dimensions in addition to showing the sloping bottom and the coordinates system that is used.
Coordinatexaxis is along the horizontal direction. Coordinate y is perpendicular to the side wall, and zis the vertical coordinate, conventionally established as positive if oriented upward from the still water line. The origin (x, z) = (0, 0) is at the intersection of the beach slope and the still water level. With this convention, it must be noted that horizontal distances measured along the flume relative to the SWL will be negative for position towards the wave maker.
Figure 3.1: Aerial view of the laboratory flume where measurements were performed. Waves generated near the bottom left corner propagate along the flume towards the top right corner.
Figure 3.3 shows measurement stations at which DCIV measurements were made and their location from the SWL mark on the beach. Waves start to overturn and break in window 11a (not shown), so DCIV measurements were made at stations marked 11b to 13b. These stations are about 0.50 m apart. An average water depth of 61.8 cm was maintained in the constant depth section of the flume. Table 3.1 shows the location of the center of each of the five stations relative to the SWL mark and the depth of the flume bed from the still water line. Most of the turbulence results that will be presented were measured at station 12a, whose center was located -2.38 m from the SWL mark on the beach and had a still water depth ofd=12.0 cm.