Aerodynamics for Airplanes

Aerodynamics for Airplanes

Hydrostatics p z 8 7 9 0 1 6 5 4 Prof. Fred Remer University of North Dakota 2 3 Air Parcel

g Reading Hess Chapter 6 pp 75 80 Wallace & Hobbs pp 67 72 Bohren & Albrecht pp 54 59

Prof. Fred Remer University of North Dakota Objectives Be able to write the vertical equation of motion Be able to state the assumptions made for hydrostatic balance Be able to describe hydrostatic balance Prof. Fred Remer University of North Dakota Objectives Be able to derive the hydrostatic

equation from the vertical equation of motion Be able to provide the definition of geopotential Be able to calculate geopotential height given geopotential Prof. Fred Remer University of North Dakota Objectives Be able to perform calculations using the hypsometric equation Be able to describe the relationship between average temperature in a layer and geopotential height Be able to describe the analysis

performed on constant pressure charts Prof. Fred Remer University of North Dakota Objectives Be able to describe the relationship between pressure and height on constant pressure charts Be able to explain the reason for the slope of pressure surfaces from the equator to the poles Prof. Fred Remer University of North Dakota

Objectives Be able to provide the definition of thickness Be able to describe the relationship between average temperature in a layer and thickness Prof. Fred Remer University of North Dakota Objectives Be able to calculate thickness given the average temperature of a layer Be able to perform calculate the average temperature of a layer given the thickness

Prof. Fred Remer University of North Dakota Vertical Equation of Motion Which forces are most important in the vertical? Coriolis Friction Pressure Gradient Gravity Prof. Fred Remer University of North Dakota

Coriolis Force Mostly Horizontal PGF NP Co Prof. Fred Remer University of North Dakota Frictional Forces Friction Mostly Horizontal

Friction Prof. Fred Remer University of North Dakota Wind Pressure Gradient Change in pressure over a given distance p x Prof. Fred Remer University of North Dakota

Pressure Gradient Three Dimensional Pressure Gradient p p p p i j k x y z Prof. Fred Remer University of North Dakota

Pressure Gradient Lets evaluate the vertical & horizontal pressure gradient 8 mb 150 mi Prof. Fred Remer University of North Dakota 500 mb 3 mi Pressure Gradient Force Vertical Pressure

Cartesian (z) z p p k z Prof. Fred Remer University of North Dakota Gravity Gravity Ability of objects to attract each other Gravity

Prof. Fred Remer University of North Dakota Gravity Gravitational Force Function of mass of each object Inversely proportional to distance GME F 2 m r Prof. Fred Remer University of North Dakota r

m ME Gravity GME g 2 r F mg G = 6.67 x 10-11 Nm2kg-2 Gravitational Acceleration (g)

Varies with Mass Radius Earths Height Above Ground Prof. Fred Remer University of North Dakota Gravity Gravitational Acceleration Assumed Constant g = 9.8 meters/sec2 = 32 ft/sec2

Variation must be accounted Prof. Fred Remer University of North Dakota Equation of Motion Vertical Equation of Motion dV 1 p a g dt z

Prof. Fred Remer University of North Dakota Equation of Motion Vertical Acceleration Important Consideration in Thunderstorms dV 1 p a g dt z Prof. Fred Remer

University of North Dakota Equation of Motion Vertical Acceleration Not So Important in Synoptic Meteorology dV 1 p a g dt z Prof. Fred Remer University of North Dakota

Vertical Equation of Motion 1 p 0 g z Vertical Gravity Pressure Gradient Only Two Forces Vertical Pressure Gradient Gravity Prof. Fred Remer University of North Dakota

Vertical Equation of Motion 1 p g z Vertical Pressure Gradient Gravity Vertical Pressure Gradient is equal to Gravity! Prof. Fred Remer University of North Dakota Vertical Equation of Motion

The Vertical Pressure Gradient is Balanced by Gravity! 1 p g z Prof. Fred Remer University of North Dakota p z Air Parcel g

z Hydrostatic Equation This relationship is known as the Hydrostatic Equation 1 p g z p z Air Parcel

g p g z Prof. Fred Remer University of North Dakota z Hydrostatic Equation Hydro - fluid Static - not moving Balance! 1 p

g z Prof. Fred Remer University of North Dakota p z Air Parcel g z Hydrostatic Equation

Rearrange a few terms 1 p g z p gz p = change in pressure z = change in height Prof. Fred Remer University of North Dakota = density g = gravity Geopotential () d p gz The potential energy of a unit mass

relative to sea level Numerically equal to the work that would be done in lifting the unit mass from sea level to the height at which the mass is located Prof. Fred Remer University of North Dakota Geopotential () d p gz The work that must be done against the Earths gravitational field in order to raise a mass of 1 kg from sea level to that point

Glossary of Meteorology Prof. Fred Remer University of North Dakota Geopotential () d p gz g Prof. Fred Remer University of North Dakota z Geopotential () z

(z) gz 0 g = f(z) g Prof. Fred Remer University of North Dakota z Geopotential Height (Z) (z) 1 Z go

go z gz 0 g = f(z) go = 9.8ms-2 The height of a given point in the atmosphere in units proportional to the potential energy of unit mass (geopotential) at this height relative to sea level Glossary of Meteorology

Prof. Fred Remer University of North Dakota Geopotential Height (Z) (z) 1 Z go go z gz 0 g = f(z) go = 9.8ms-2

Used in upper air calculations Small difference between height (z) and geopotential height (Z) in lower atmosphere Prof. Fred Remer University of North Dakota Hydrostatic Equation We dont normally measure density. p g z

Im too young to die! Eliminate density. Prof. Fred Remer University of North Dakota Hydrostatic Equation Using the Ideal Gas Law p R dTv

p R dTv Substitute into Hydrostatic Equation p g z Prof. Fred Remer University of North Dakota p p g z

R dTv Hydrostatic Equation p p g z R dTv Rearrange terms p gz R dTv p Prof. Fred Remer University of North Dakota

Hydrostatic Equation p gz R dTv p Remember ... Substitute d gz p d gz R dTv p

Prof. Fred Remer University of North Dakota Hydrostatic Equation p d gz R dTv p Integrate between two pressure levels p2 2 1 R d p1 Prof. Fred Remer

University of North Dakota p Tv p Hydrostatic Equation p2 2 1 R d p1 p Tv p

Divide both sides by go and reverse the limits 2 1 Rd go go Prof. Fred Remer University of North Dakota p p2 Tv p p1 Hydrostatic Equation 2 1 Rd

go go Remember Z go Rd Z 2 Z1 go Prof. Fred Remer University of North Dakota p p2 Tv p

p1 Substitute! p p2 Tv p p1 Hypsometric Equation Rd Z 2 Z1 go p p2 Tv p p1

The height difference between two pressure surfaces depends on Virtual Temperature Pressure Prof. Fred Remer University of North Dakota Hypsometric Equation Rd Z 2 Z1 go At Sea Level p1 = sea level pressure Z1 = 0 m

Rd Z2 go Prof. Fred Remer University of North Dakota p p2 Tv p p1 p SL p2 p

Tv p Hypsometric Equation Rd Z2 go p SL p2 p Tv p

Z2 p2= 700 mb Height of a pressure surface p1= SLP Prof. Fred Remer University of North Dakota Constant Pressure Chart Height of a pressure surface Prof. Fred Remer

University of North Dakota Constant Pressure Chart Contours - lines of constant height Prof. Fred Remer University of North Dakota Constant Pressure Chart Height of a pressure surface Function of Temperature 700 mb

Rd Z2 go p SL p2 p Tv p H z = 3120 m

L z = 2850 m SL Prof. Fred Remer University of North Dakota Constant Pressure Chart How does height compare to pressure? 690 mb z p H

L 700 mb 710 mb 720 mb 10,000 ft SL Prof. Fred Remer University of North Dakota Constant Pressure Chart 690 mb 700 mb H

L p 710 mb 10,000 ft 720 mb SL Prof. Fred Remer University of North Dakota Constant Pressure Chart High Height High Pressure

Low Height Low Pressure 700 mb H z = 3120 m L z = 2850 m SL Prof. Fred Remer University of North Dakota Hypsometric Equation

Hypsometric Equation Relates the distance between pressure surfaces z2 Rd Z 2 Z1 go Prof. Fred Remer University of North Dakota p p2 Tv p p1 p2= 500

mb 5480 m z1 60 m p1= 1000 mb Thickness Thickness (Z) Distance between pressure surfaces Z Z 2 Z 1 Rd

Z go p p2 Tv p p1 z2 p2= 500 mb Z = Z2 - Z1 p = p2 - p1 z1 p1= 1000 mb

5480 m 60 m Prof. Fred Remer University of North Dakota Thickness Calculate Thickness Rd Z go p

p2 Tv p p1 Integrate!? Problem Temperature varies with height Prof. Fred Remer University of North Dakota Thickness What are we going to do? Prof. Fred Remer University of North Dakota

Thickness Two methods 1.) New-Miracle Analysis 2.) Fudge Prof. Fred Remer University of North Dakota Thickness Take the average temperature of the layer Prof. Fred Remer

University of North Dakota Thickness Substitute average temperature Mathematically .... p1 p Rd Z T v p2 p go p p2 Tv p Tv p 1 p

p2 p p1 where weighted average Prof. Fred Remer University of North Dakota Thickness Simplify p1 p Rd Z T v

p2 p go p1 Rd Z T v d(ln p) p2 go p1 Rd Z T v ln go p2 Prof. Fred Remer University of North Dakota Hypsometric Equation

p1 Rd Z T v ln go p2 Z = height between pressure surfaces p1 = lower pressure surface p2 = upper pressure surface Tv = average temperature in layer Rd = Gas Constant go = gravity Prof. Fred Remer University of North Dakota Hypsometric Equation p1 Rd Z T v ln

go p2 Pressure decreases logarithmically Prof. Fred Remer University of North Dakota Hypsometric Equation Pressure decreases logarithmically Prof. Fred Remer University of North Dakota

Hypsometric Equation p1 Rd Z T v ln go p2 Thickness of a layer depends on temperature Prof. Fred Remer University of North Dakota Hypsometric Equation Thickness depends on temperature

500 mb Warm Z Cold Z 1000 mb Prof. Fred Remer University of North Dakota Hypsometric Equation

Thickness depends on temperature Decreasing Pressure b b m m 0 0 30 40 Cold Air North

Prof. Fred Remer University of North Dakota 0 50 m b Warm Air South 0 0

6 b m 0 70 m b Hypsometric Equation Prof. Fred Remer University of North Dakota Thickness

p1 Rd Z T v ln go p2 1000-500 mb common 287Jkg 1K 1 1000 Z T v ln 2 9.8ms 500 Z 20.3mT v

Prof. Fred Remer University of North Dakota Thickness Prof. Fred Remer University of North Dakota Prof. Fred Remer University of North Dakota

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