| dc.description.abstract | The problem of flow past disturbances in circular pipes has wide engineering applications. Most available information pertains to turbulent approach flow. In this thesis, the effects of disturbances in circular pipes are studied with fully developed laminar approach flow.
The disturbances investigated include several kinds of orifices and perforated plates. The study aims to:
Understand the development of pressure and velocity fields downstream of the disturbance.
Determine loss coefficient and settling length for various disturbance geometries.
Flow Characteristics
The Hagen–Poiseuille approach flow, existing a few diameters upstream, gets distorted by the disturbance.
Downstream, the distorted flow gradually redevelops into Hagen–Poiseuille flow.
Immediately downstream, the flow may be laminar or turbulent, depending on:
Reynolds number
Geometry of the disturbance
The problem is broadly classified into two categories:
Reynolds number below critical value ? Developing flow remains laminar.
Reynolds number above critical value ? Developing flow becomes relaminarising type.
Analytical Approach
Numerical solutions of boundary layer equations for entrance flow show that laminar flow development can be described by a family of exponent-type profiles:
u=um(1?ra)nu = u_m \left(1 - \frac{r}{a}\right)^nu=um?(1?ar?)n
where:
uuu = axial velocity
umu_mum? = mean axial velocity
rrr = radial coordinate
aaa = pipe radius
nnn = exponent (ranges from 2 to ?)
Results are presented in terms of these profiles, facilitating study of any profile in the family.
Approximate solutions based on momentum and energy integral equations are obtained and compared with numerical solutions.
Experimental Investigations
Conducted for 31 disturbances, including:
Sharp-edged concentric, eccentric, and segmental orifices
Quadrant-edged orifices
Long orifices
Perforated plates
p ratio (p=flow area through disturbancepipe areap = \frac{\text{flow area through disturbance}}{\text{pipe area}}p=pipe areaflow area through disturbance?) varied from 0.2 to 0.8.
Pressure and velocity measurements:
33 diameters upstream
333 diameters downstream
Two oils used as fluid medium.
806 runs for pressure profiles at Reynolds numbers R=5R = 5R=5 to 200020002000.
Velocity profiles observed at R?1000R \approx 1000R?1000 and additional observations at two more Reynolds numbers for some disturbances.
Loss Coefficient
Defined as:
K=heu22gK = \frac{h_e}{\frac{u^2}{2g}}K=2gu2?he??
where:
KKK = loss coefficient
heh_ehe? = excess head loss due to disturbance
Key Findings:
Variation of KKK with RRR discussed in detail.
For sharp-edged concentric orifices:
A critical Reynolds number exists where KKK attains a minimum.
Interpreted as the Reynolds number at which turbulence originates downstream.
For disturbances with large ppp ratios:
KKK curve shows sharp transition.
For disturbances with significant friction losses (e.g., long orifices):
No critical Reynolds number observed.
At very low Reynolds numbers:
K?1RK \propto \frac{1}{R}K?R1?
The Reynolds number below which this linear relationship holds is determined for all disturbances.
Empirical and Theoretical Models
Interpolation formula developed for determining KKK for ppp ratios near known values.
Loss coefficients obtained for nominal ppp ratios: 0.2, 0.4, 0.6, 0.8.
Comparative study of:
Magnitude of KKK
Variation with geometry, edge radius, eccentricity, orifice length, and number of perforations.
Semi-empirical theory based on continuity and momentum equations:
Predicts KKK using discharge coefficient.
Works well for low/moderate ppp ratios (p?0.4p \leq 0.4p?0.4) and Reynolds numbers above critical value.
For long orifices:
Combined method with entrance flow model gives theoretical solution for KKK.
Applicable for low ppp ratios common in practice.Two methods are proposed to determine the critical Reynolds number at which turbulence originates downstream of the disturbance:
Method 1: Based on a study of the loss coefficient variation.
Method 2: Based on the length of the pressure recovery region downstream of the disturbance.
A pressure index is proposed to facilitate determination of the critical Reynolds number.
Values of the critical Reynolds number obtained by these two methods are compared. With guidance from existing flow visualization studies, a plausible explanation is given for the difference between the two sets of values.
The significance of the critical Reynolds number is clearly demonstrated through a study of the development of pressure and velocity fields.
Pressure and Velocity Field Development
Based on experimental observations, a detailed study is made on the variation of pressure and velocity fields in the developing region.
For disturbances where velocity profiles in the developing region are axisymmetric, the family of exponent-type profiles effectively describes the developing profiles even for Reynolds numbers above the critical value, with n values determined experimentally.
Based on pressure measurements, the variation of settling length with Reynolds number is obtained for all disturbances:
For R?100R \leq 100R?100, settling length increases linearly with R.
Studies include variations of:
Centreline velocity
Energy coefficient
Momentum coefficient
Settling lengths based on these criteria are compared with those obtained from pressure measurements.
Influence of Turbulence
For Reynolds numbers well above the critical value, the influence of turbulence on flow development is discussed by comparing:
Actual rate of flow development
Rate in the absence of turbulence (obtained from numerical solutions of boundary layer equations)
Using mean flow data for pressure and velocity in the energy integral equation, approximate estimates are obtained for the distance of turbulence decay.
Conclusions
Important conclusions are listed separately.
Guidelines for future research are provided. | |
