Shear stress fluctuation

Above relation (1) between cr and y is exact in linear response, where nonlinear contributions in 7 are neglected in the stress. The linear response modulus (to be denoted as g (f)) itself is defined in the quiescent system and describes the small shear-stress fluctuations always present in thermal equilibrium [1, 3]. Often, oscillatory deformations at fixed frequency co are applied and the frequency dependent storage- (G (m)) and loss- (G"((u)) shear moduli are measured in or out of phase, respectively. The former captures elastic while the latter captures dissipative contributions. Both moduli result from Fourier-transformations of the linear response shear modulus g (f), and are thus connected via Kramers-Kronig relations. 

In experimental aerodynamics, the surface hot wire probe has proved to be the most successful standard measurement technique to determine the laminar-to-turbulent flow transition, local separation, and shear stress fluctuations. The flush-mounted thermal shear stress sensor is one of the most successful techniques for shear stress measurement and is available in various forms, i.e., sensor skin, etc. [4], due to the rapid development of MEMS manufacturing processes. 

Frequency spectra of wall shear stress fluctuations and of velocity fluctuations indicate relative attenuation of the high frequency components (Figure 3). Measurements in boundary-free flows have also usually found that the small-scale components are suppressed relative to the large-scale. 

Figure 3. Spectral density function for wall shear stress fluctuations for solutions of Separan. Dashed line solvent. Closed circles 34.6% drag reduction, Re = 27,600. Closed squares 64.5% drag reduction. Re = 57,600. Solid line 62.9% drag reduction. Re = 10,000. 

CORRELATION FOR THE DYNAMIC COEFFICIENT C, 338 Amplitude and Phase of Shear Stress Fluctuation, 341... 

Attempts to explore this complicated interaction and to model the response of the eddy viscosity and turbulent shear stresses to the time variation of pressure gradients in turbulent air flow over a solid wavy surface have been made by Thorsness et al. [85] and Abrams and Hanratty [89]. Large variations of the amplitude and phase angle of the surface shear stress with the dimensionless wave number were predicted (Figure 3). The analysis shows that the surface shear stress fluctuation is shifted upstream with respect to the wave elevation and the phase shift varies in the range of 0-80° in comparison to the constant phase shift predicted by Benjamin [84],... 

Equations 29-31 can be used to express the shear stress fluctuation in terms of its amplitude and phase ... 

The frequency spectra of wall shear stress fluctuations for drag-reducing flows show the relative attenuation of high frequency components. The decrease in high frequency content and increase in low frequency are observed in the frequency spectra of axial as well as transverse fluctuations for drag-reducing flows. 

The fatigue evaluation methodology is based on the minimum shear stress theory of failure. This consists of finding the amplitude (one half of the range) through which the maximum shear stress fluctuates. This is obtained in the ASME Code procedure by using stress difference and stress intensities (twice the maximum shear stress). 

With turbulent flow, shear stress also results from the behavior of transient random eddies, including large-scale eddies which decay to small eddies or fluctuations. The scale of the large eddies depends on equipment size. On the other hand, the scale of small eddies, which dissipate energy primarily through viscous shear, is almost independent of agitator and tank size. 

In turbulent motion, the presence of circulating or eddy currents brings about a much-increased exchange of momentum in all three directions of the stream flow, and these eddies are responsible for the random fluctuations in velocity The high rate of transfer in turbulent flow is accompanied by a much higher shear stress for a given velocity gradient. 

In streamline flow, E is very small and approaches zero, so that xj p determines the shear stress. In turbulent flow, E is negligible at the wall and increases very rapidly with distance from the wall. LAUFER(7), using very small hot-wire anemometers, measured the velocity fluctuations and gave a valuable account of the structure of turbulent flow. In the operations of mass, heat, and momentum transfer, the transfer has to be effected through the laminar layer near the wall, and it is here that the greatest resistance to transfer lies. 

Fig. 19—Shear stress and chain angle as a function of sliding distance, from simulations of alkanethiolates on Au(111) at temperature 1 K (a) results from commensurate sliding show a stick-slip motion with a period of 2.5 A, (b) in incommensurate case both shear stress and chain angle exhibit random fluctuations with a much smaller average friction [45],...

Although the transport properties, conductivity, and viscosity can be obtained quantitatively from fluctuations in a system at equilibrium in the absence of any driving forces, it is most common to determine the values from experiments in which a flux is induced by an external stress. In the case of viscous flow, the shear viscosity r is the proportionality constant connecting the magnitude of shear stress S to the flux of matter relative to a stationary surface. If the flux is measured as a velocity gradient, then... 

Velocity fluctuations can also cause extra apparent shear stress components. An element of fluid with a non-zero velocity component in the x-direction possesses an x-component of momentum. If this element of fluid also has a non-zero velocity component in the y-direction then as it moves in the y-direction it carries with it the x-component of momentum. The mass flow rate across a plane of area 8x8z normal to the y-coordinate direction is pvy8x8z and the x-component of momentum per unit mass is vx, so the rate of transfer of x-momentum in they-direction is given by the expression... 

In general, the time-averaged value of the product of the fluctuations is non-zero so there is an additional flux of x-momentum in the y-direction due to the velocity fluctuations v x and v y. This momentum flux is equivalent to an extra apparent shear stress acting in the x-direction on the plane normal to the y-coordinate direction. Consequently, the mean total shear stress for turbulent flow can be written as... 

In equation 1.94, (Tyx)v is the viscous shear stress due to the mean velocity gradient dvjdy and pv yv x is the extra shear stress due to the velocity fluctuations v x and v y. These extra stress components arising from the velocity fluctuations are known as Reynolds stresses. (Note that if the positive sign convention for stresses were used, the sign of the Reynolds stress would be negative in equation 1.94.)... 

As the fluid s velocity must be zero at the solid surface, the velocity fluctuations must be zero there. In the region very close to the solid boundary, ie the viscous sublayer, the velocity fluctuations are very small and the shear stress is almost entirely the viscous stress. Similarly, transport of heat and mass is due to molecular processes, the turbulent contribution being negligible. In contrast, in the outer part of the turbulent boundary layer turbulent fluctuations are dominant, as they are in the free stream outside the boundary layer. In the buffer or generation zone, turbulent and molecular processes are of comparable importance. 

From equation 1.41, the total shear stress varies linearly from a maximum fw at the wall to zero at the centre of the pipe. As the wall is approached, the turbulent component of the shear stress tends to zero, that is the whole of the shear stress is due to the viscous component at the wall. The turbulent contribution increases rapidly with distance from the wall and is the dominant component at all locations except in the wall region. Both components of the mean shear stress necessarily decline to zero at the centre-line. (The mean velocity gradient is zero at the centre so the mean viscous shear stress must be zero, but in addition the velocity fluctuations are uncorrelated so the turbulent component must be zero.)... 

In the amorphous pattern we observe a spatial fluctuation of density as well as different repulsive and attractive forces. Describing the amorphous structure with (n — m)/n = x and using the 6—12-potential of Eq. (5) we obtain an internal shearing stress... 

When a fluid is in turbulent flow past a rigid surface, fluctuations of velocity in the direction normal to the surface are inhibited, and very close to the surface they may he negligible. Then the Reynolds shear stress is small compared with the viscous stresses, and it has been common to describe the region as a laminar sublayer. In fact, turbulent fluctuations of velocity in planes parallel to the wall are considerable in comparison with the mean velocity. 

In the wall thickness fluctuations up to 5 % may occur. As a result of the uneven temperature in the molten polymer during rotation, and also by the not always exactly reproducible rate of cooling, deviations in the dimensions of the finished product may amount to 5 %. Requirements are, that the materials can be molten completely, that the melt is sufficiently low-viscous, and that the molten polymer does not degrade too rapidly. Besides plasticised PVC, HDPE and LDPE are often used, as well as copolymers of PE such as EVA (ethylene - vinyl acetate copolymerj.Because the shear stresses in this process are extremely low, a narrow molar mass distribution is to be recommended, as discussed in 5.4. Cycle times vary between 3 and 40 minutes, dependent on the wall thickness. Cycle times can be reduced considerably by using machines with multiple moulds, since the cycle time... 

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu 2, -pv 2, and -pw 2 are always non-zero because they contain squared velocity fluctuations. The shear stresses -puV, -pu w, -pv w and are associated with correlations between different velocity components. If, for instance, u and v were statistically independent fluctuations, the time average of their product uV would be zero. However, the turbulent stresses are also non-zero and are usually large compared to the viscous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

In Table IV, we see that established techniques for velocity measurement allow us to determine the average momentum flux, average velocity, turbulent intensities, and shear stress. Next on the list, to complete the flow field description, is the fluctuation mass flux, and first on the combustion field list is the temperature and major species densities of the flame gases. 

Thus, the extra terms arising from the fluctuating velocity components are such that their effects are the same as an increase in the viscous shear stress and it is for this reason that they are termed the turbulent or Reynolds stress terms. 

The fluctuations give rise to a turbulent-shear stress which may be analyzed by referring to Fig. 5-12. 

For a unit area of the plane P-P, the instantaneous turbulent mass-transport rate across the plane is pv. Associated with this mass transport is a change in the x component of velocity u. The net momentum flux per unit area, in the x direction, represents the turbulent-shear stress at the plane P-P, or pv u When a turbulent lump moves upward (v > 0), it enters a region of higher u and is therefore likely to effect a slowing-down fluctuation in u , that is, u < 0. A similar argument can be made for v < 0, so that the average turbulent-shear stress will be given as... 

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