this is my own research

1. IMPACT OF GRAVITY ON FLUID MECHANICS
MODELS
 Aamir Raza 2018-ag-7885 BSc. Agri. ENGG.
 Muhammad Shazaib 2018-ag-7886 section:A2

2. Contents
1. Abstract
2. Introduction
3. Methodology
4. Results
5. Conclusions

3. 1. ABSTRACT
 We used Mathematical formulae to describe fluid mechanics models
 In this formulae gravity is taken as constant.
 Actually, gravity is not constant.
 it is changing depending on mass distribution into the body of the Earth, mass density,
altitude and topography.
 This research is focused on the gravity influence on the different hydraulics models .
 And fluid mechanic formulae in order to point out
 That gravity acceleration should not be treated routinely as “constant”.

4. 2. INTRODUCTION
 Earth gravity field covers all around the planet.
 Almost in all hydraulic formulae which describe hydraulic structures
 we take gravity as constant.
 Its value is often adopted without accurate determination for certain area.
 In this way error of gravity acceleration occur and result were not too significant.
 Gravity acceleration is changes with the passage of time.

5. 2. INTRODUCTION
 Variation of gravity depending on distribution of masses inside Earth.
 And Earth’ position respective to Sun and Moon.
 Earth gravity field is changeable with time and that could not be treated as constant.
 Aforementioned reasons have a consequence that Earths’ gravity field is not constant in
space and time.

6. 2. INTRODUCTION
 This fact could be simply expressed by formulae:
g=g(x,y,z,t)

𝜕𝑔
𝜕𝑥
≠0 ;
𝜕𝑔
𝜕𝑦
≠ 0 ;
𝜕𝑔
𝜕𝑧
≠ 0 ;
𝜕𝑔
𝜕𝑡
≠ 0 ……………..(1)
where:
g= gravity
x=x- coordinate in Cartesian 3D World coordinate system
y=y-coordinate in Cartesian 3D World coordinate system
z=z-coordinate in Cartesian 3D World coordinate system
t=time
 Gravity is mostly the field of two scientific disciplines 1) physical geodesy 2) geophysics
 From the aspect of physical geodesy the earth gravity field is reached with aim to determine geoid
 From the aspect of geophysics the main aim of gravity field determination is to find out the Earths’ interior.

7. 2. INTRODUCTION
Geoid

8. 3. METHODOLOGY
 Hydraulics models and hydraulic structures are described by formulae.
 These Models are based on numerical and empirical research.
 All these researches are based on measurements which contain unavoidable errors.
 Errors propagation through certain hydraulic model is depending
 on the form of formulae which describe observed hydraulic phenomenon.
 Formulae which describe hydraulic models often contain some coefficients.
 These coefficients were determined through empirical research.
 These coefficients represent values obtained from limited sample and under certain
conditions.
 Empirical coefficients also are rounded which means that rounding error can exists
and influence output quantity.

9. 3. METHODOLOGY
 Mathematically it could be expressed in following way:
𝑒,
≠ 0 ……………(2)
Or
e𝜖[𝑒 − ∆𝑒 , 𝑒 + ∆𝑒 ] …………..(3)
 Where e is empirical coefficient, 𝑒,is first derivative and ∆𝑒is possible deviation of empirical
coefficient.
 Error propagation also depends on the shape of the formula.
 That describes it and on initial conditions i.e. Measured or adopted values of parameters.
 The error propagation shall not be the same for quadratic and logarithmic function.

10. 3. METHODOLOGY
 For following analysis a mathematical models will be used. Considering function dependent of n
arguments
𝛹 = 𝛹 𝑥1. 𝑥2, 𝑥3 … … … . 𝑥𝑛 …………………….(4)
 To approximate its value in initial point increased for its increment defined by first order derivative:
𝛹 ≈ 𝛹( 𝑥1
0
, 𝑥2
0
, 𝑥3
0
… … 𝑥n
0)+
𝑖=𝐼
𝑛
𝜕𝜓
𝜕𝑋
∆𝑋𝑖 ………………..(5)
 From formula follows that first order derivative reads
Δ𝛹 =
𝑖=𝐼
𝑛
𝜕𝜓
𝜕𝑋
∆𝑋𝑖 …………….(6)
 According to low of error propagation the root mean square error for formula shall read:
𝑚Δ𝛹 =
𝑖=𝐼
𝑛
(
𝜕𝜓
𝜕𝑋
∆𝑋𝑖)
2
…………………..(7)

11. 3. METHODOLOGY
 where m∆𝑋𝑖are the root mean square errors of increments ∆𝑋𝑖.
 the formula can be written as
Δ𝜓 = 𝑛𝛿 …………………(8)
 Where
𝛿 =
𝜕𝜓
𝜕𝑋
∆𝑋1 =
𝜕𝜓
𝜕𝑋
∆𝑋2 … … . =
𝜕𝜓
𝜕𝑋
∆𝑋 𝑛 …………………..(9)
Bearing in mind (6), (8) and (9) immediately follows:
Δ 𝑋 𝑖
=
Δ𝜓
𝑛
1
𝜕𝛹
𝜕𝑥ⅈ
……………………(10)
 The formula for normal gravity is given by the means of conventional series:
𝛾 = 9.780327(1+0.0053024sin 𝜙2
- 0.0000058sin 2𝜙2
)𝑚𝑠−2
………………….(11)
 where normal gravity is denoted by 𝛾 and altitude is denoted by ∅.

12. 3. METHODOLOGY
 Formula (8) has an accuracy of 1𝜇𝑚𝑠−2
=0.1 mGal.
 The normal gravity 𝛾 belongs to the interval of
(9.780327 𝑚𝑠−2 − 9.832186𝑚𝑠−2) when 𝜙𝜖[0,90°].
 Average of normal gravity over ellipsoid is 𝛾 =9.797𝑚𝑠−2
 According to literature the extreme values of gravity acceleration are
 9.76392𝑚𝑠−2 at Huascaran, Peru (𝜙=-9.12°, o=-77.60°) minimum value
 And 9.83366 𝑚𝑠−2
at Arctic Sea (𝜙 =86.71 °, o=61.29°) maximum value
 it means that variation range of gravity acceleration on Earth is about 0.07𝑚𝑠−2

13. 3. METHODOLOGY
Huascaran, Peru

14. 3. METHODOLOGY ARCTIC SEA

15. 4. RESULTS
 In this research, models for bed shear stress and ogee spillway are performed
according to described methodology.
 Model for bed shear stress is:
𝝉 = 𝝆𝒈𝒉𝒍
 𝜏= bed shear stress;
 𝜌 = density of water;
 h= - water depth and
 l=slope of the water surface.
 Applying formula (5) on formula (7) we get:
𝝉=𝜌0 𝑔0ℎ0 𝐼0 + ∆𝜌0 𝑔0ℎ0 𝐼0 + 𝜌0∆𝑔0ℎ0 𝐼0 + 𝜌0 𝑔0∆ℎ0 𝐼0+𝜌0 𝑔0ℎ0∆𝐼0

16. 4. RESULTS
 Increment of function 𝝉 due to increment of arguments (or their errors) has following form:
 When initial values and limit increment for ∆ 𝝉 are given
 it is possible to determine intervals for every argument’s increment
 For given values of bed shear stress the maximum values of uncertainty are shown in table 1.

17. 4. RESULTS
 On the base of results gravity is smaller than its real variation.
 That implies that there exist cases when the gravity acceleration could not be treated as “constant”
 Model for ogee-spillway is:
 Increment of discharge function which is consequence of arguments’ errors reads:

18. 4. RESULTS

19. 5. CONCLUSIONS
 The impact of gravity acceleration participates in numerous models of hydraulic
and structures
 But it is usually considered as a constant because variation of gravity on Earth is
about 0.07𝑚𝑠−2
 Gravity acceleration, however, is not constant Because it depends on numerous
factors which also change with time.
 In this research a few examples for bed shear stress and ogee-spillway models
were considered.
 And it is shown there are cases for hydraulic models and structures where needed
variation of gravity for obtaining given models’ variation
 total increment of function is smaller than real variation of gravity acceleration.
 These cases suggest that gravity acceleration shall not be routinely treated as
“constant”.

20. 5. CONCLUSIONS
 Availability of data about gravity acceleration which justify attitude,
 That every hydraulic model or structure shall be provided with adequate data
 Gravity acceleration for geographic location where certain hydraulic model or structure is located.
 Changes of gravity acceleration in time justifies its measurement
 Because hydraulic models and structures are assumed to last for decades.
 By Increasing accuracy of impact of gravity acceleration in hydraulic model
 Make it possible to decrease the influence of other influences in hydraulic models.

21. RESEARCH BY

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