Steady gradually varied flow in channels

Ustálené nerovnoměrné proudění v korytech I. Teorie No. 12203-0904/KOSICE03

Steady gradually varied flow in channels I. Theory Ing. Hana Uhmannová, CSc., Vysoké učení technické v Brně, Fakulta stavební, Ústav vodních staveb Brno University of Technology, Faculty of Civil Engineering, Institut of Water Structures This project has been funded with support from the European Commission. This presentation/publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Úvod – Introduce  The flow of water with free surface presents some of the most common problems of fluid mechanics.  Natural channels, artificial (constructed) channels and floods provide a number of important problems which have led to development the theories and computations methods in hydraulics of open channel.  The main characteristic of many studies is the computation of the water surface profiles for specific channels with varying hydraulic characteristics for steady, non-uniform gradually varied flow.  In most cases it is possible to treat the problem as an one-dimensional flow and a number of simple and powerful methods have been developed.

Typy otevřených koryt – Types of open channels In open channels water flows with a free surface at atmospheric pressure. Depending on the character of their formation is divided into: • natural river - channel,

• artificial (constructed) channel.

Upravené koryta

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Artificial channels

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These channels are constructed or developed by human effort. They include for examples: • irrigation cannals, • navigation channels, • power canals, • drainage ditches, etc. – They are usually constructed in a regular cross-section shape. – Analysis of flow in such well defined channels will give reasonably accurate results. A channel in which the cross sectional shape, size and the bottom slope are constant is termed as prismatic channel. Most of the artificial channels are prismatic channels over long stretches.

Přirozené řeky –

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Natural channels

Natural channels include all watercourses that exist naturally on the earth. Natural channels can be very different. They are not regular, non-prismatic and their materials of construction can very widely. The surface roughness will often change with time, distance and even elevation. Analysis of flow is usually more difficult than for artificial channels. The situation may be further complicated if the boundary is not fixed i.e. erosion and deposition of sediments.

All natural channels generally have varying cross section and they are non-prismatic.

Typy proudění

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Types of flow

Time is the criterion.

Steady flow -

velocity at any cross section does not vary in magnitude or direction with time, depth of flow at a particular points does not change or can be considered constant for the time interval under consideration.

Unsteady flow -

velocity varies with time and position, depth changes with time.

In most open channel flow problems, flow conditions are studied under steady flow conditions only.

Flood flows in rivers and rapidly varying surges in canals are some examples of unsteady flow.

Typy proudění

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Types of flow

Space is the criterion.

Uniform flow -

depth of flow and the mean velocities are the same at successive cross sections in any reach, Uniform flow may only occur in a channel with a constant cross section.

Non-uniform flow -

the flow properties vary along the channel, occurs in reaches with uniform or varying cross sections affected by other controls or its own changing shape, which produce accelerated flow or backwater,

A prismatic channel carrying a certain discharge with a constant velocity is an example of uniform flow.

Typy proudění

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Types of flow

The non-uniform flow can be classified as gradually varied flow and rapidly varied flow.

Gradually varied flow -

change of depth is gradual so that the curvature of streamlines is not excessive, loss of energy is essentially due to boundary friction. Therefore, the distribution of pressure in the vertical direction may be taken as hydrostatic.

Rapidly varied flow -

curvature in a varied flow is large, and the depth changes appreciably over short lengths.

Ustálené nerovnoměrné proudění v přirozených a upravených korytech Steady non-uniform flow in natural and artificial channels -

The steady, uniform flow and the steady, non-uniform flow are the most fundamental types of flow used in engineering hydraulics. The steady uniform flow rarely occurs in natural rivers. Natural channels usually are non-prismatic and steady state flow is nonuniform.

Steady state flow implies that the discharge Q through each cross section of the channel is the same but  Y (depth),  A (cross sectional area)  and v (mean velocity) usually depend on x (position).

Teorie – Theory Equations for Basic Profile Calculation  Open-channel flow with a free surface where atmospheric pressure is assumed.  The cross-sectional area of open-channel flow is not known.  Elevation of the free surface depends on the discharge and may change with position. In the following, it is assumed that: -

the flow can be modelled by a 1D approach, the bed of the channel is rigid, the fluid is incompressible, the pressure distribution in the water body is hydrostatic (pressure increases linearly with depth).

Water surface profiles are computed from one cross section to the next by solving the Energy equation (Bernoulli equation) with an iterative

procedure called the standart step method.

Teorie – Theory Equations for Basic Profile Calculation – STANDART STEP METHOD The Energy equation from one cross section to the next

is writen as follows : Where:

Z1, Z2 …elevation of the main channel inverts Y1, Y2 …depth of water at cross section V1, V2 …average velocities α1, α2 …velocity weighting coefficients g

…gravitational acceleration

he

…energy head loss

HEC-RAS Hydraulic Reference Manual CHAPTER 2

Teorie – Theory The energy head loss - he - between two cross sections is comprised of friction losses and contration or expansion losses

The equatin for energy head loss is follows : Where: L

… discharge weighted reach length

S f … representative friction slope between two section C … expansion or contraction loss coefficient

The distance weighted reach lenght L is calculated as: Where: L lob , Lch , L rob

… cross section reach lengths specified for flow in the left overbank (lob), main channel (ch), and right overbank (rob), respectively

Q lob + Qch + Q rob … arithmetic average of the flows between sections for the left overbank, main channel, and right overbank, respectively HEC-RAS Hydraulic Reference Manual - CHAPTER 2

Teorie – Theory Cross Section Subdivision for Conveyace Calculations 

The determination of total conveyance and the velocity coefficient for a cross section requires that flow be subdivided into units for which the velocity is uniformly distributed.



Conveyance is calculated within each subdivision from the following form of Manning´s equation (based on English units) :

HEC-RAS Hydraulic Reference Manual CHAPTER 2

Where: Q

… dischardge

K

… conveyance for subdivision

n

… Manning´s roughness coefficient for subdivision

A … flow area for subdivision R … hydraulic radius for subdivision ( A / P

area/wetted perimeter )

Teorie – Theory Cross Section Subdivision for Conveyace Calculations  The approach used in HEC-RAS is to subdivide flow in the overbank areas using the input cross section n-value break points (locations where n-value change) as the basic for subdivision (Figure 2 – 2).

HEC-RAS Hydraulic Reference Manual CHAPTER 2

 The program sumps up all the incremental conveyances in the overbanks to obtain a conveyance for the LEFT overbank and RIGHT overbank.

 The MAIN channel conveynce is normally computed as a single conveyance element.  The TOTAL conveynce for the cross section is obtained by summing the tree subdivision conveyances (left, channel, and right).

Teorie – Theory Composite Manning´s n for the Main Channel

HEC-RAS Hydraulic Reference Manual CHAPTER 2

Where:

nc

… composite or equivalent coefficient of roughness

P

… wetted perimeter of entire main channel

Pi

… wetted perimeter of subdivision I

ni

… coefficient of roughness for subdivision

Teorie – Theory The calculation water surface profiles are dependent on the flow regime as follows: Flow is subcritical -

boundary condition is entered into the lower profile, the direction of computation is upstream.

Flow is supercritical -

boundary condition is entered into the upper profile, the direction of computation is downstream.

The direction of computation

Modelování proudění v otevřených korytech Modeling Flow in Open Natural and Constructed Channels Modelování je jedna z metod poznávání a má historii.

Modeling is one of the methods of cognition and has an history.

Člověk se snažil popsat věci a jevy (které se nacházely v jeho prostředí) nejrůznějšími prostředky:  verbálním popisem,  graficky,  pomocí matematické symboliky,  fyzikálně a technicky realizovanými modely.

Human tried to describe things and phenomena (that are present in its environment) by various means :  verbal description ,  graphically,  using mathematical symbols,  physically and technically implemented models.

Produktem jeho úvah byly určité systémy -

modely vyjadřovaly a znázorňovaly podstatné vlastnosti

reálných systémů – objektů.

The product of his reasoning were certain systems

models

describing the essential characteristics of real systems - objects.

Modelování – Modeling Modelování je jedna z metod poznávání a má historii.

Modeling is one of the methods of cognition and has an history.

Člověk se snažil popsat věci a jevy (které se nacházely v jeho prostředí) nejrůznějšími prostředky:  verbálním popisem,  graficky,  pomocí matematické symboliky,  fyzikálně a technicky realizovanými modely.

Human tried to describe things and phenomena (that are present in its environment) by various means :  verbal description ,  graphically,  using mathematical symbols,  physically and technically implemented models.

Produktem jeho úvah byly určité systémy -

The product of his reasoning were certain systems

modely

vyjadřovaly a znázorňovaly podstatné vlastnosti

reálných systémů – objektů.

models describing the essential characteristics of real systems - objects.

Modelování – Modeling V inženýrské praxi vždy existuje dvojice

model – reálný systém.

In engineering practice there is always a pair

model - a real system .

Z tohoto hlediska rozdělujeme modely na dvě skupiny:

From this perspective, models are divided into two groups:

Modely umožňující analyzovat reálný systém,

Models allowing to analyze the real system ,

(t.j. konkretizovat a zpřesňovat naše představy o už existujícím systému. Jsou to např. modely atomu, biologických struktur, ale i počítačů, či jiných zařízení).

(ie, specify and refine our ideas about the already existing system. These include the models of the atom, biological structures , as well as computers or other devices).

Modely, které vznikly v důsledku projektování a návrhu.

Models that have arisen as a result of the planning and design.

(Realizace modelovaných zařízení se teprve plánuje. Jde o úlohy syntézy, projektování a konstruování. Činnost bývá podporovaná výpočetní technikou).

(The implementation of modeled plants are being planned. It is the job of synthesis, design and construction. The activity is supported by computer technology).

Definice – Definition Model je účelově zjednodušené zobrazení nějakého reálného nebo abstraktního světa. Matematický model je popis určitého výseku vnějšího světa matematickými prostředky. Je to obecnější druh modelu, který se dělí na:    

abstraktní, ideově matematický model – výpočtový algoritmus, exaktní teoretické řešení – výpočtový model dané úlohy, konkrétní, předmětový matematický model, kterým může být nastavené řešení na počítači.

Model is purposefully simplified representation of some real or abstract world. The mathematical model is a description of a certain section of the outside world by mathematical means. It's kind of a more general model, which is divided into:    

abstract mathematical model - computational algorithm, exact theoretical solution - computational model of the task, specific mathematical model, which can be set on the computer solution .

Modelování – Modeling Rozdělení - Distribution : fyzikální modely physical models

- hydraulický - hydraulic - aerodynamický - aerodynamic - jiný analogový - other analog

matematické modely mathematical models

- stanovení cílů - setting goals - analýza problému a matematická formulace, problem analysis and mathematical formulation - zjednodušující předpoklady (1D, 2D), simplifying assumptions (1D, 2D) - výběr a aplikace metody řešení (diferenční,variační…), selection and application of methods for solving (differential , variation ...) - algoritmizace úlohy a sestavení zdrojového programu, algorithm and formation of the source program - numerické testy - numerical tests - kalibrace - calibration - citlivostní analýza - sensitivity analysis - verifikace modelu - verification of the model - simulace vybraných stavů - simulation of selected situation

Modelování – Modeling Matematické modely proudění tekutin lze rozdělit dle: Mathematical models of fluid flow can be divided according to: hydraulického režimu proudění hydraulic flow regime časového režimu proudění time flow regimes

účelu, kterému slouží the purpose it serves

povahy vstupních veličin character of input variables

- laminární - laminar - turbulentní - turbulent

- stacionární, statické - stationary , static - nestacionární, dynamické - non-stationary, dynamic

- simulační - simulation - řídící - control - optimalizační - optimization

- stochastické - stochastic - deterministické - deterministic

Modelování – Modeling Matematické modely proudění tekutin lze rozdělit dle: Mathematical models of fluid flow can be divided according to: charakteristik kapaliny the characteristics fluid

teplotních podmínek temperature conditions

prostorové dimenze spatial dimensions

-s homogenní nebo nehomogenní kapalinou - of homogeneous or inhomogeneous fluid -se stlačitelnou nebo nestlačitelnou kapalinou - the compressible and incompressible fluid - s newtonskou nebo nenewtonskou kapalinou - a Newtonian fluid or nenewtonskou

- izotermické – isothermal - neizotermické - non-isothermal

- jednorozměrné - one-dimensional – 1D - dvourozměrné - two-dimensional – 2D - trojrozměrné - three-dimensional – 3D

Design, performance and operation of selected water structures

Děkuji za pozornost Thank you for your attention

Kosice, 2012