Blasius flow

The constants A and B can be determined from the solutions available for the simple cases in which either = 0 or uw = 0. The first limiting case corresponds to the dissolution of a planar wall in a semiinfinite liquid in steady flow (Blasius flow), while the second limit corresponds to steady-state diffusion with chemical reaction in a stagnant fluid. For Blasius flow at large Schmidt numbers [2],... 

Expressions for the Nusselt number for zero suction are available for various values of m [4], For the Blasius flow (m = 0), the expression has the form... 

One may note that the constant for the Blasius flow is 0.664 instead of 0.88. If the velocity at infinity Ut 0, then... 

This result has the following consequence for the completeness of basis function constructed from the eigenvectors obtained by stability analysis of external flows. It has been clearly shown by Mack (1976) that internal flows, like the channel flow, has denumerable infinite number of eigen modes and any arbitrary applied disturbance can be expressed in terms of this complete basis set. However, for external flows, as we have seen for the Blasius flow in Table 2.1 that there are only a few discrete eigenmodes and it is not possible to express any arbitrary functions in terms of these only, in the absence of any other singularities for this flow. 

Gilpin, R.R., Imura, H. and Cheng, K.G. (1978). Experiments on the onset of longitudinal vortices in horizontal Blasius flow heated from below, ASME J. Heat Transfer 100 71-77. 

Wu, R.S. and Cheng, K.C. (1976). Thermal instability of Blasius flow along horizontal plates, Int. J. Heat Mass Transfer 19 907-913. 

Let us investigate steady-state convective diffusion on the surface of a flat plate in a longitudinal translational flow of a viscous incompressible fluid at high Reynolds numbers (the Blasius flow). We assume that mass transfer is accompanied by a volume reaction. In the diffusion boundary layer approximation, the concentration distribution is described by the equation... 

R. R. Gilpin, H. Imura, and K. C. Cheng, Experiments on the Onset of Longitudinal Vortices in Horizontal Blasius Flow Heated From Below, J. Heat Transfer (100) 71-77,1978. 

R. S. Wu and K. C. Cheng, Thermal Instability of Blasius Flow Along Horizontal Plates, Int. J. Heat Mass Transfer (19) 907-913,1976. 

It appears that our measurement follows the Smith model. But it is noted that the Reynolds number of our data for low Gortler parameter is considerably small. Then, it seems that the Blasius flow no longer holds good for those data. In connection with this problem Ragab and Nayfeh computed the neutral stability curves taking into account of the effect of displacement thickness of boundary layer. According to their results the curves approach to that of the Smith model departing from that of modified Smith model at low wave number when R becomes small. 

For turbulent flow in smooth tubes, the Blasius equation gives the friction facdor accurately for a wide range of Reynolds numbers. 

Yooi24) has proposed a simple modification to the Blasius equation for turbulent flow in a pipe, which gives values of the friction factor accurate to within about 10 per cent. The friction factor is expressed in terms of the Metzner and Reed(I8) generalised Reynolds number ReMR and the power-law index n. 

The right-hand side of equation 10.224 gives numerical values which are very close to those obtained from the Blasius equation for the friction factor (j> for the turbulent flow of a fluid through a smooth pipe at Reynolds numbers up to about 106. 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

A simple approximate form of the relation between u+ and y+ for the turbulent flow of a fluid in a pipe of circular cross-section may be obtained using the Prandtl one-seventh power law and the Blasius equation. These two equations have been shown (Section 11.4) to be mutually consistent. 

The Blasius relation between friction factor and Reynolds number for turbulent flow is ... 

For hydrodynamically smooth pipes, through which fluid is flowing under turbulent conditions, the shear stress is given approximately by the Blasius equation ... 

For flow in a smooth pipe, the friction factor for turbulent flow is given approximately by the Blasius equation and is proportional to the Reynolds number (and hence the velocity) raised to a power of -2. From equations 12.102 and 12.103, therefore, the heat and mass transfer coefficients are both proportional to w 75. 

In the above equations, a is a coefficient with the value of 1.0 for single phase flow and 2.0 for multi-phase flow [6], and pis an adjustable coefficient and has a value of 2.1 by fitting the experimental results for the two phase flow. The flow resistant coefficient is determined by the Blasius equation. 

By substituting the well-known Blasius relation for the friction factor, Eq. (45) in Table VII results. Van Shaw et al. (V2) tested this relation by limiting-current measurements on short pipe sections, and found that the Re and (L/d) dependences were in accord with theory. The mass-transfer rates obtained averaged 7% lower than predicted, but in a later publication this was traced to incorrect flow rate calibration. Iribame et al. (110) showed that the Leveque relation is also valid for turbulent mass transfer in falling films, as long as the developing mass-transfer condition is fulfilled (generally expressed as L+ < 103) while Re > 103. The fundamental importance of the Leveque equation for the interpretation of microelectrode measurements is discussed at an earlier point. 

Forced convection, burning of a flat plate. This classical exact analysis following the well-known Blasius solution for incompressible flow was done by Emmons in 1956 [7], It includes both variable density and viscosity. Glassman [8] presents a functional fit to the Emmons solution as... 

Considerable effort has been expended in trying to And algebraic expressions to relate/to Re and eld,. For turbulent flow in smooth pipes, the simplest expression is the Blasius equation ... 

This is consistent with the Blasius type of expression used for the friction factors in deriving the Martinelli parameter. Using the value n = 0.20 and expressing the ratio of flow rates in terms of the quality... 

The Bernoulli equation can now be written for the liquid in channel flow in the bottom part of the tube, and for the liquid in slug flow in the upper part. The acceleration terms are then neglected, and the friction factors for each type of liquid flow found from the Blasius equation and from true Reynolds numbers. The resulting equations cannot be readily evaluated because of the two hydraulic-radius terms involved in the two types of flow, and an unknown fraction defining the relative mass of liquid in each part of the tube. 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

Similarity is perhaps best know in the context of external boundary-layer flow, such as the Blasius solution (cf., the books by Schlichting [350] or White [429]). In these cases an independent-variable transformation is found in which a single new independent variable is a special combination of the physical spatial coordinates. In this book we are generally more concerned with internal flows where the approaches to finding similarity can differ. 

Many of the phenomena of the boundary layer are explainable on the basis of the theory advanced by Prandtl al the University of Gottingen laboratory nearly half a century ago. In the same flow-research group were others, like Blasius, who broadened and experimentally confirmed the original hypotheses. 

F or turbulent pipe flow, the friction velocity u = Vx ,/p used earlier in describing the universal turbulent velocity profile may be used as an estimate for V Together with the Blasius equation for the friction factor from which e may be obtained (Eq. 6-214), this provides an estimate for the energy-containing eddy size in turbulent pipe flow ... 

For turbulent flow of Newtonian fluids in smooth pipes, two common correlations are those of Blasius [413] for 3000 < Re < 100 000 ... 

The relation between cost per unit length C of a pipeline installation and its diameter d is given by C = a + bd where a and b are independent of pipe size. Annual charges are a fraction of the capital cost. Obtain an expression for the optimum pipe diameter on a minimum cost basis for a fluid of density p and viscosity p flowing at a mass rate of G. Assume that the fluid is in turbulent flow and that the Blasius equation is applicable, that is the friction factor is proportional to the Reynolds number to the power of minus one quarter. Indicate clearly how the optimum diameter depends on flowrate and fluid properties. 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

The flow profiles with H > 2.591, correspond to velocity distributions with inflection point and these are the decelerated flows or flows with adverse pressure gradient. On the contrary, the flow profiles with H < 2.591, correspond to - < 0 (the accelerated flows). The figure with = 0 and H = 2.59, corresponds to the Blasius profile. The profile with j3h = I and H = 2.22 corresponds to the stagnation point flow. The other two profiles in Fig. 2.7 are for flows with adverse pressure gradient and the crosses on the profile indicate the locations of the inflexion point. The profile for /3h = —0.1988 H = 4.032) corresponds to the case of incipient separation. 

Presented solution once again, demonstrates the far-field to correspond to the TS mode obtained by linear stability analysis. For Re = 1000 and LVo = 0.1, the calculated impulse response displays TS wave with ar = 0.279826 and a = —0.007287. The results are shown at a height of y = 1.205(5 - the location of the outer maximum of the eigenvector. Considering the stability properties of the Blasius profile, one expects the flow to be stable for Re = 400 and 4000 - with the latter case showing higher damping than the former, as clearly seen in Fig. 2.21. 

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